MTH120 College Algebra Chapter 6.2

 6.2 Graphs of Exponential Functions

Graphing exponential functions is a common task in mathematics, and it helps visualize how these functions behave over time or for different values of the independent variable. To graph an exponential function, follow these steps:

1. Determine the Form of the Exponential Function: Exponential functions typically have the form $�(�)=�\cdot {�}^{�}$, where $�$ is the initial value at $�=0$, and $�$ is the base of the exponential growth or decay.

2. Identify Key Parameters:

   • Determine the values of $�$ and $�$. $�$ is the initial quantity or value, and $�$ represents the rate of growth or decay.
   • Decide the range of values for $�$ that you want to graph.

3. Create a Table of Values:

   • Choose several values of $�$ within your desired range.
   • Calculate the corresponding $�(�)$ values using the formula. This will give you points to plot on the graph.

4. Plot the Points:

   • Use the values from your table to plot points on the coordinate plane, where the x-axis represents $�$ and the y-axis represents $�(�)$.

5. Draw the Exponential Curve:

   • Connect the plotted points with a smooth curve. The curve will be an increasing curve for exponential growth or a decreasing curve for exponential decay.

6. Label the Axes and Graph:

   • Label the x-axis and y-axis appropriately. Include units if applicable.
   • Title the graph to indicate the function or context.

7. Optional Additions:

   • If you have asymptotes (horizontal or vertical), plot them as dashed lines.
   • If you're dealing with a real-world problem, provide context for the graph.

Here's an example:

Graph the function $�(�)=2\cdot 3^{�}$ for $�$ in the range -2 to 2.

1. Function form: $�(�)=2\cdot 3^{�}$.

2. Parameters: $�=2$, $�=3$.

3. Table of values:

   $�$	$�(�)$
   -2	1/18
   -1	2/3
   0	2
   1	6
   2	18

4. Plot the points and draw the exponential curve.

5. Label the axes: x-axis as "x" and y-axis as "f(x)." Title the graph if needed.

6. Your graph should show the exponential growth curve of $�(�)$ as it increases when $�$ increases.

Remember that exponential functions can either grow rapidly (with a base greater than 1) or decay rapidly (with a base between 0 and 1). The specific values of $�$ and $�$ and the range you choose for $�$ will determine the shape and behavior of the graph.

The parent function $�(�)={�}^{�}$ represents a basic exponential function. Understanding its characteristics is essential for understanding the behavior of exponential functions as a whole. Here are the key characteristics of the graph of the parent exponential function $�(�)={�}^{�}$:

1. Exponential Growth or Decay:

   • The graph can represent either exponential growth or decay, depending on the value of the base $�$.
   • If $0<�<1$, the function represents exponential decay. As $�$ increases, $�(�)$ gets smaller and approaches zero.
   • If $�>1$, the function represents exponential growth. As $�$ increases, $�(�)$ gets larger and approaches infinity.

2. Y-Intercept:

   • The parent function $�(�)={�}^{�}$ passes through the point $(0,1)$ on the graph, regardless of the value of $�$.

3. Domain and Range:

   • The domain of the function is all real numbers ($-\mathrm{\infty }<�<\mathrm{\infty }$).
   • The range of the function depends on the value of $�$. For exponential growth, the range is $(0,\mathrm{\infty })$, and for exponential decay, it is $(0,1)$.

4. Asymptote:

   • The x-axis ($�=0$) acts as a horizontal asymptote for exponential growth (as $�$ approaches negative infinity) or for exponential decay (as $�$ approaches positive infinity).

5. Symmetry:

   • The graph is never symmetric. It is always increasing or decreasing.

6. Rate of Change:

   • The rate of change of the function increases as you move right along the x-axis for exponential growth and decreases for exponential decay.
   • The slope of the tangent line at any point is proportional to the function's value at that point.

7. Intercepts:

   • The function has no x-intercepts, except at $�=0$.
   • The y-intercept is $(0,1)$ for all values of $�$.

8. Behavior Near the Origin:

   • For values of $�>1$, the function starts very close to the origin and grows rapidly.
   • For values of $0<�<1$, the function starts very close to the origin and decreases rapidly.

9. Exponential Function Examples:

   • For $�=2$, the function represents exponential growth.
   • For $�=1\mathrm{/}2$, the function represents exponential decay.

10. Increasing or Decreasing Intervals:

    • The function is always either increasing or decreasing over its entire domain.

Understanding these characteristics is fundamental when dealing with exponential functions. The specific behavior of the graph will depend on the value of $�$ and how it affects the rate of growth or decay.

Sketching the graph of an exponential function of the form $�(�)={�}^{�}$ is a straightforward process once you understand the basic characteristics of such functions. Here are the steps to sketch the graph:

1. Determine Exponential Growth or Decay:

   • Identify whether the function represents exponential growth or decay based on the value of $�$:
     • If $0<�<1$, it's exponential decay.
     • If $�>1$, it's exponential growth.

2. Find Key Points:

   • Choose a few values of $�$ to calculate corresponding $�(�)$ values. For simplicity, you can choose $�=-1,0,1$ to get a sense of the behavior.

3. Calculate $�(�)$ Values:

   • Use the formula $�(�)={�}^{�}$ to calculate the $�(�)$ values. For example, if you chose $�=-1,0,1$, calculate $�(-1)$, $�(0)$, and $�(1)$.

4. Plot Key Points:

   • Plot the points on the graph with $�$ on the horizontal axis and $�(�)$ on the vertical axis.

5. Determine the Behavior:

   • Since exponential functions never cross the x-axis except at $(0,1)$, you know the graph will never touch or go below the x-axis.

6. Asymptotes:

   • For exponential growth ($�>1$), the x-axis ($�=0$) serves as a horizontal asymptote as $�$ approaches negative infinity.
   • For exponential decay ($0<�<1$), the x-axis serves as a horizontal asymptote as $�$ approaches positive infinity.

7. Connect the Dots:

   • Draw a smooth curve that passes through the points and follows the behavior described in steps 5 and 6.

8. Label Axes and Add Context (if needed):

   • Label the x-axis as "x" and the y-axis as "f(x)."
   • Provide a title or context for the graph, if applicable.

9. Specify the Domain and Range:

   • Mention that the domain is all real numbers ($-\mathrm{\infty }<�<\mathrm{\infty }$).
   • Mention that the range is either $(0,\mathrm{\infty })$ for exponential growth or $(0,1)$ for exponential decay.

10. Scale the Axes:

    • Choose appropriate scales for the x and y-axes to make the graph clear and easily readable. The scales should be chosen to fit the range of the function.

Here are a couple of examples:

Example 1: Exponential Growth (b = 2)

• Determine that it's exponential growth (because $�=2$).
• Find key points: $(0,1)$, $(-1,0.5)$, $(1,2)$.
• Plot these points, and draw a curve that increases rapidly to the right.
• Label axes and provide context if needed.

Example 2: Exponential Decay (b = 1/3)

• Determine that it's exponential decay (because $�=1\mathrm{/}3$).
• Find key points: $(0,1)$, $(-1,3)$, $(1,1\mathrm{/}3)$.
• Plot these points, and draw a curve that decreases rapidly to the right.
• Label axes and provide context if needed.

Remember that the behavior of the graph depends on the value of $�$, and the graph will never cross the x-axis except at $(0,1)$.

Graphing transformations of exponential functions involves modifying the basic exponential function $�(�)={�}^{�}$ to create new functions with shifts, stretches, and reflections. These transformations can change the shape and position of the graph. Here are the steps to graph a transformed exponential function:

1. Start with the Basic Exponential Function: Begin with the parent function $�(�)={�}^{�}$, which represents either exponential growth or decay.

2. Identify the Transformation Components:

   • Determine the following transformation components:
     • Horizontal Shift (or Translation) ($ℎ$): This value represents how the graph is shifted horizontally (left or right).
     • Vertical Shift (or Translation) ($�$): This value represents how the graph is shifted vertically (up or down).
     • Stretch or Compression ($�$): This value scales the function vertically (stretch or compress).

3. Apply the Transformations:

   • Modify the basic function based on the transformation components:
     • Horizontal shift: Replace $�$ with $�-ℎ$.
     • Vertical shift: Add $�$ to the function.
     • Stretch or compression: Multiply the function by $�$.

4. Graph the Transformed Function:

   • Use the modified function to plot the graph.
   • Choose several values of $�$ to calculate the corresponding $�(�)$ values.
   • Plot these points, and draw the graph that follows the behavior of the modified function.

5. Label Axes and Provide Context:

   • Label the x-axis and y-axis as appropriate.
   • If the function represents a real-world problem, provide context for the graph.

6. Determine Domain and Range:

   • Define the domain and range of the transformed function based on its behavior.

7. Scale the Axes:

   • Adjust the scales on the axes to make the graph clear and easily readable, considering the range of the function.

8. Asymptotes and Behavior:

   • Consider whether there are any asymptotes (horizontal or vertical) based on the transformations and the behavior of the graph.

Here's an example to illustrate these steps:

Example: Graph the transformed exponential function $�(�)=2^{�}-3$

• Transformation Components:

  • Horizontal shift: None ($ℎ=0$).
  • Vertical shift: Down 3 units ($�=-3$).
  • Stretch or compression: None ($�=1$).

• Transformation Applied:

  • Apply the vertical shift by subtracting 3 from the basic function: $�(�)=2^{�}-3$.

• Graph the Transformed Function:

  • Plot the graph by calculating several values of $�$ and the corresponding $�(�)$ values.
  • Draw a curve that follows the behavior of the modified function, shifting the basic exponential decay graph downward by 3 units.

• Label Axes and Context:

  • Label the x-axis and y-axis.
  • Provide context if the function represents a real-world problem.

• Domain and Range:

  • Define the domain and range based on the behavior of the graph.

• Scale the Axes:

  • Choose appropriate scales for the x and y-axes to make the graph clear and easily readable.

• Asymptotes and Behavior:

  • Consider whether there are any asymptotes based on the transformations and the behavior of the graph.

By following these steps and understanding the impact of each transformation component, you can effectively graph transformed exponential functions.

Graphing a vertical shift, also known as a vertical translation, involves moving the entire graph of a function vertically up or down the y-axis. This transformation is achieved by adding or subtracting a constant value to the original function. Here are the steps to graph a vertical shift:

Step 1: Start with the Original Function: Begin with the original function you want to shift vertically. For example, let's say you have the function $�(�)$.

Step 2: Identify the Vertical Shift (Translation): Determine the amount you want to shift the graph vertically. If you want to move the graph up, you add a positive constant to the function; if you want to move it down, you subtract a positive constant from the function. Let's say you want to shift the graph up by $�$ units.

Step 3: Apply the Vertical Shift: Modify the original function by adding or subtracting the constant value $�$:

• For a vertical shift upward by $�$, use $�(�)+�$.
• For a vertical shift downward by $�$, use $�(�)-�$.

So, you now have a new function $�(�)=�(�)+�$ (for an upward shift) or $�(�)=�(�)-�$ (for a downward shift).

Step 4: Calculate Key Points: To graph the shifted function $�(�)$, you can choose a few values of $�$ and calculate the corresponding $�(�)$ values. This will give you points to plot on the graph.

Step 5: Plot the Shifted Function: Plot the points on the coordinate plane, where the x-axis represents $�$ and the y-axis represents $�(�)$.

Step 6: Label Axes and Add Context (if needed): Label the x-axis as "x" and the y-axis as "g(x)." If the function represents a real-world problem, provide context for the graph.

Step 7: Specify the Domain and Range: Define the domain and range of the shifted function based on its behavior.

Step 8: Scale the Axes: Choose appropriate scales for the x and y-axes to make the graph clear and easily readable. The scales should be chosen to fit the range of the function.

For example, let's say you have the original function $�(�)={�}^2$ and you want to shift it upward by 3 units. The new function is $�(�)={�}^2+3$.

• Calculate some points for $�(�)$ by choosing values of $�$ (e.g., -2, -1, 0, 1, 2) and adding 3 to the corresponding $�(�)$ values to get $�(�)$ values.
• Plot these points and draw the shifted parabolic graph.

Remember that a vertical shift does not change the shape of the graph; it simply moves the entire graph up or down along the y-axis.

Graphing a horizontal shift, also known as a horizontal translation, involves moving the entire graph of a function horizontally along the x-axis. This transformation is achieved by adding or subtracting a constant value to the independent variable (x). Here are the steps to graph a horizontal shift:

Step 1: Start with the Original Function: Begin with the original function you want to shift horizontally. For example, let's say you have the function $�(�)$.

Step 2: Identify the Horizontal Shift (Translation): Determine the amount you want to shift the graph horizontally. If you want to move the graph to the right, you subtract a positive constant from $�$; if you want to move it to the left, you add a positive constant to $�$. Let's say you want to shift the graph to the right by $ℎ$ units.

Step 3: Apply the Horizontal Shift: Modify the original function by adding or subtracting the constant value $ℎ$ to the independent variable $�$:

• For a horizontal shift to the right by $ℎ$, use $�(�-ℎ)$.
• For a horizontal shift to the left by $ℎ$, use $�(�+ℎ)$.

So, you now have a new function $�(�)=�(�-ℎ)$ (for a rightward shift) or $�(�)=�(�+ℎ)$ (for a leftward shift).

Step 4: Calculate Key Points: To graph the shifted function $�(�)$, you can choose a few values of $�$ and calculate the corresponding $�(�)$ values. This will give you points to plot on the graph.

Step 5: Plot the Shifted Function: Plot the points on the coordinate plane, where the x-axis represents $�$ and the y-axis represents $�(�)$.

Step 6: Label Axes and Add Context (if needed): Label the x-axis as "x" and the y-axis as "g(x)." If the function represents a real-world problem, provide context for the graph.

Step 7: Specify the Domain and Range: Define the domain and range of the shifted function based on its behavior.

Step 8: Scale the Axes: Choose appropriate scales for the x and y-axes to make the graph clear and easily readable. The scales should be chosen to fit the range of the function.

For example, let's say you have the original function $�(�)={�}^2$ and you want to shift it to the right by 2 units. The new function is $�(�)=�(�-2)$.

• Calculate some points for $�(�)$ by choosing values of $�$ (e.g., -2, -1, 0, 1, 2) and substituting $�-2$ into the original function to get $�(�)$ values.
• Plot these points and draw the shifted parabolic graph.

Remember that a horizontal shift does not change the shape of the graph; it simply moves the entire graph horizontally along the x-axis.

The parent function $�(�)=��$ represents a basic linear function. Understanding how shifts, or translations, work for this parent function is crucial in graphing and analyzing linear functions. Shifts involve moving the entire graph horizontally (a horizontal shift) or vertically (a vertical shift). Here are examples of each type of shift:

1. Vertical Shift (Translation):

• Original Function: $�(�)=2�$

  Vertical Shift Up by 3 Units:

  • To shift the graph of $�(�)=2�$ upward by 3 units, you add 3 to the function:

  New Function: $�(�)=2�+3$

  • This shifts the entire graph upward without changing the slope. The original function passes through the point $(0,0)$, while the shifted function $�(�)$ passes through $(0,3)$. The slope remains 2.

  Vertical Shift Down by 2 Units:

  • To shift the graph of $�(�)=2�$ downward by 2 units, you subtract 2 from the function:

  New Function: $ℎ(�)=2�-2$

  • This shifts the entire graph downward without changing the slope. The original function passes through $(0,0)$, while the shifted function $ℎ(�)$ passes through $(0,-2)$. The slope remains 2.

2. Horizontal Shift (Translation):

• Original Function: $�(�)=2�$

  Horizontal Shift to the Right by 3 Units:

  • To shift the graph of $�(�)=2�$ to the right by 3 units, you subtract 3 from $�$ within the function:

  New Function: $�(�)=2(�-3)$

  • This shifts the entire graph to the right without changing the slope. The original function passes through $(0,0)$, while the shifted function $�(�)$ passes through $(3,0)$. The slope remains 2.

  Horizontal Shift to the Left by 1 Unit:

  • To shift the graph of $�(�)=2�$ to the left by 1 unit, you add 1 to $�$ within the function:

  New Function: $�(�)=2(�+1)$

  • This shifts the entire graph to the left without changing the slope. The original function passes through $(0,0)$, while the shifted function $�(�)$ passes through $(-1,0)$. The slope remains 2.

These examples illustrate how vertical and horizontal shifts impact the graph of the parent function $�(�)=��$. Vertical shifts move the graph up or down, while horizontal shifts move it left or right. The slope (in this case, 2) remains the same in all these examples.

If you have an equation in the form $�(�)=��+�+�$ and you want to approximate its solution or graph it using a graphing calculator, you can follow these steps.

Let's use an example equation for this demonstration: $�(�)=2�+3+1$. In this equation, $�=2$, $�=3$, and $�=1$.

1. Turn On Your Graphing Calculator: Ensure your graphing calculator is powered on and ready to use.

2. Access the Graphing Function:

   • Most graphing calculators have a dedicated "Graph" or "Y=" button. Press this button to access the graphing function.

3. Enter the Equation: You'll need to enter the equation $�(�)=2�+3+1$ into your calculator. You typically do this by:

   • Pressing the "Y=" button to access the function input.
   • Entering "2x + 3 + 1" into one of the available function slots.

4. Adjust the Window Settings (Optional): Depending on your calculator's default window settings, you may need to adjust the viewing window to see the graph clearly. This typically involves setting the x-min, x-max, y-min, and y-max values to encompass the region you're interested in.

5. Graph the Equation: After entering the equation and adjusting the window settings, press the "Graph" button or an equivalent button on your calculator.

6. Interpret the Graph:

   • The graph of the equation will be displayed on the calculator's screen.
   • You can use the calculator's arrow keys to move the viewing window around and explore the graph.

7. Finding Solutions (Intercepts):

   • To find solutions (x-intercepts or roots), you can use the calculator's built-in features. For example, to find where the graph crosses the x-axis (i.e., the solutions), you can use the "Zero" or "Root" feature.
   • Alternatively, you can analyze the graph visually to approximate where it crosses the x-axis.

8. Saving and Storing:

   • Most graphing calculators allow you to store graphs or equations for future use. Check your calculator's manual for instructions on saving or storing functions.

These are the general steps to graph an equation of the form $�(�)=��+�+�$ using a graphing calculator. The specific steps may vary depending on the brand and model of your calculator. Refer to your calculator's manual for detailed instructions on graphing and finding solutions using your particular calculator.

Graphing a stretch or compression involves changing the slope of a linear function, which is represented by the equation $�(�)=��$. A stretch makes the function steeper (increasing the slope), while a compression makes it less steep (decreasing the slope). Here are the steps to graph a stretched or compressed linear function:

Step 1: Start with the Original Linear Function: Begin with the parent function $�(�)=��$, where $�$ is the slope of the line. The slope represents the rate at which $�$ changes with respect to $�$.

Step 2: Identify the Stretch or Compression Factor: Determine the stretch or compression factor, denoted as $�$. This factor will change the slope of the linear function. If $�>1$, it represents a stretch (increase in slope), and if $0<�<1$, it represents a compression (decrease in slope). Let's say you have $�=2$ for a stretch or $�=1\mathrm{/}2$ for a compression.

Step 3: Apply the Stretch or Compression: Modify the original function by multiplying the slope ($�$) by the stretch or compression factor ($�$):

• For a stretch, use $�(�)=�\cdot ��$.
• For a compression, use $�(�)=�\cdot ��$.

So, you now have a new function $�(�)$ for a stretched or compressed linear function:

• For a stretch, $�(�)=2�$ if $�=2$.
• For a compression, $�(�)=\frac{1}{2}�$ if $�=\frac{1}{2}$.

Step 4: Calculate Key Points: To graph the stretched or compressed function $�(�)$, you can choose a few values of $�$ and calculate the corresponding $�(�)$ values. This will give you points to plot on the graph.

Step 5: Plot the Stretched or Compressed Function: Plot the points on the coordinate plane, where the x-axis represents $�$ and the y-axis represents $�(�)$.

Step 6: Label Axes and Provide Context (if needed): Label the x-axis as "x" and the y-axis as "g(x)." If the function represents a real-world problem, provide context for the graph.

Step 7: Specify the Domain and Range: Define the domain and range of the stretched or compressed function based on its behavior.

Step 8: Scale the Axes: Choose appropriate scales for the x and y-axes to make the graph clear and easily readable. The scales should be chosen to fit the range of the function.

For example, let's say you have the original function $�(�)=2�$ and you want to stretch it by a factor of 3. The new function is $�(�)=3\cdot 2�=6�$.

• Calculate some points for $�(�)$ by choosing values of $�$ (e.g., -2, -1, 0, 1, 2) and multiplying 6 by the corresponding $�(�)$ values to get $�(�)$ values.
• Plot these points and draw the stretched linear graph.

Remember that a stretch or compression changes the slope of the graph while keeping the direction of the line (positive or negative) the same.

Let's look at some specific examples of stretched and compressed linear functions and how they are graphed. We'll consider two cases, one for a stretch and one for a compression.

Example 1: Stretch (Increase in Slope)

Original Function: $�(�)=2�$

Stretch Factor: $�=3$ (a stretch by a factor of 3)

New Function: $�(�)=3\cdot 2�=6�$

• In this example, the original function $�(�)=2�$ is stretched by a factor of 3, resulting in the new function $�(�)=6�$. This means the slope of the line has increased.

• Calculate points for the stretched function $�(�)$:

  • When $�=-1$, $�(-1)=6\cdot (-1)=-6$
  • When $�=0$, $�(0)=6\cdot (0)=0$
  • When $�=1$, $�(1)=6\cdot (1)=6$

• Plot these points and draw the stretched linear graph. The original function $�(�)=2�$ is less steep compared to the stretched function (g(x) = 6x).

Example 2: Compression (Decrease in Slope)

Original Function: $�(�)=3�$

Compression Factor: $�=\frac{1}{2}$ (a compression by a factor of 1/2)

New Function: $ℎ(�)=\frac{1}{2}\cdot 3�=\frac{3}{2}�$

• In this example, the original function $�(�)=3�$ is compressed by a factor of $1\mathrm{/}2$, resulting in the new function $ℎ(�)=\frac{3}{2}�$. This means the slope of the line has decreased.

• Calculate points for the compressed function $ℎ(�)$:

  • When $�=-1$, $ℎ(-1)=\frac{3}{2}\cdot (-1)=-\frac{3}{2}$
  • When $�=0$, $ℎ(0)=\frac{3}{2}\cdot (0)=0$
  • When $�=1$, $ℎ(1)=\frac{3}{2}\cdot (1)=\frac{3}{2}$

• Plot these points and draw the compressed linear graph. The original function $�(�)=3�$ is steeper compared to the compressed function $ℎ(�)=\frac{3}{2}�$.

These examples demonstrate how a stretch or compression changes the slope of a linear function while keeping the direction of the line (positive or negative) the same. A stretch increases the slope, while a compression decreases it. The graphical representation of these changes is evident in the steepness of the lines on the graph.

Stretching and compressing the parent function $�(�)=��$ involves altering the slope of the linear function. The parent function $�(�)=��$ is a straight line with a slope of $�$. When you apply stretches or compressions, you are essentially modifying the slope ($�$) of the line. Here's a summary of how stretches and compressions affect the parent function:

1. Stretching the Parent Function:

• A stretch occurs when you increase the slope ($�$) of the parent function. Mathematically, you multiply $�$ by a value greater than 1 (e.g., $�$ becomes $2�$ or $3�$).
• As you increase the slope ($�$), the line becomes steeper. It means that for each unit increase in $�$, the corresponding $�$ value increases more. The line is more "vertical."
• For example, if the parent function is $�(�)=2�$, a stretch would be $�(�)=3�$. This results in a line that's steeper than the original.

2. Compression of the Parent Function:

• A compression occurs when you decrease the slope ($�$) of the parent function. Mathematically, you multiply $�$ by a value between 0 and 1 (e.g., $�$ becomes $\frac{1}{2}�$ or $\frac{1}{3}�$).
• When the slope is decreased, the line becomes less steep. It means that for each unit increase in $�$, the corresponding $�$ value increases less. The line is less "vertical."
• For example, if the parent function is $�(�)=2�$, a compression would be $�(�)=\frac{1}{2}�$. This results in a line that's less steep than the original.

In summary, stretches and compressions of the parent function $�(�)=��$ are all about changing the slope ($�$) of the linear function. A stretch makes the line steeper by increasing $�$, while a compression makes the line less steep by decreasing $�$. These transformations are fundamental in graphing and understanding linear functions and their behavior.

Graphing reflections involves flipping a function over an axis or line, creating a mirror image of the original graph. Two common types of reflections are vertical reflections and horizontal reflections. Here are the steps to graph reflections:

1. Start with the Original Function: Begin with the original function that you want to reflect. For example, let's consider the function $�(�)$.

2. Identify the Axis or Line of Reflection: Determine the axis or line over which you want to reflect the graph. Common choices are the x-axis for vertical reflections and the y-axis for horizontal reflections. Let's say you want to perform a vertical reflection over the x-axis.

3. Apply the Reflection:

• For a vertical reflection over the x-axis, replace $�(�)$ with $-�(�)$.
• For a horizontal reflection over the y-axis, replace $�$ with $-�$ within the function.

Let's illustrate both types of reflections with examples:

Vertical Reflection (Over the X-Axis):

• Original Function: $�(�)={�}^2$
• Vertical Reflection: $�(�)=-{�}^2$

Horizontal Reflection (Over the Y-Axis):

• Original Function: $�(�)={�}^2$
• Horizontal Reflection: $ℎ(�)=(-�{)}^2={�}^2$

4. Calculate Key Points: To graph the reflected function, calculate key points. This often includes using symmetry to generate points on one side of the axis of reflection and then reflecting them across the axis to get points on the other side.

5. Plot the Reflected Function: Plot the points on the coordinate plane, where the x-axis represents $�$ and the y-axis represents the reflected $�(�)$.

6. Label Axes and Provide Context (if needed): Label the x-axis as "x" and the y-axis as "f(x)" or the reflected "f(x)." If the function represents a real-world problem, provide context for the graph.

7. Specify the Domain and Range: Define the domain and range of the reflected function based on its behavior. Note that the domain and range may change depending on the reflection.

8. Scale the Axes: Choose appropriate scales for the x and y-axes to make the graph clear and easily readable. The scales should be chosen to fit the range of the function.

These steps will help you graph reflections over the x-axis or y-axis. The choice of axis or line of reflection determines the type of reflection you perform. The reflected graph will be a mirror image of the original graph over the chosen axis.

Translations of the exponential function involve shifting the graph horizontally or vertically. Here's a summary of how translations impact the standard exponential function $�(�)={�}^{�}$:

Vertical Translations:

1. Vertical Shift Up (or Down):

   • Transformation: $�(�)\to �(�)+�$ or $�(�)\to �(�)-�$
   • Effect: Shifts the entire graph vertically by $�$ units.
   • Equation Example: If $�(�)=2^{�}$, a vertical shift up by 3 units would be $�(�)=2^{�}+3$.

2. Vertical Stretch (or Compression):

   • Transformation: $�(�)\to �\cdot �(�)$ or $�(�)\to \frac{1}{�}\cdot �(�)$ (where $�>1$ for stretch, $0<�<1$ for compression)
   • Effect: Changes the steepness of the graph vertically.
   • Equation Example: If $�(�)=2^{�}$, a vertical stretch by a factor of 3 would be $�(�)=3\cdot 2^{�}$.

Horizontal Translations:

1. Horizontal Shift Right (or Left):

   • Transformation: $�(�)\to �(�-ℎ)$ or $�(�)\to �(�+ℎ)$
   • Effect: Shifts the entire graph horizontally by $ℎ$ units.
   • Equation Example: If $�(�)=2^{�}$, a horizontal shift left by 2 units would be $�(�)=2^{�+2}$.

2. Horizontal Stretch (or Compression):

   • Transformation: $�(�)\to �(��)$ or $�(�)\to �\left(\frac{�}{�}\right)$ (where $�>1$ for compression, $0<�<1$ for stretch)
   • Effect: Changes the steepness of the graph horizontally.
   • Equation Example: If $�(�)=2^{�}$, a horizontal compression by a factor of 2 would be $�(�)=2^{�\mathrm{/}2}$.

Combined Translations:

You can combine vertical and horizontal translations in a single exponential function. For example, $�(�)=�\cdot {�}^{��+�}+�$ involves vertical and horizontal shifts and stretches.

Summary:

• Vertical Translations: Alter the y-values. Upward shifts by adding a constant, downward shifts by subtracting a constant, stretches or compressions by multiplying by a constant.

• Horizontal Translations: Alter the x-values. Rightward shifts by subtracting a constant from $�$, leftward shifts by adding a constant to $�$, stretches or compressions by multiplying or dividing $�$ by a constant.

Understanding these translations is crucial for accurately graphing and analyzing exponential functions in various contexts.

Algebraic 3. The graph of f(x)=3x is reflected about the y-axis and stretched vertically by a factor of 3. What is the equation of the new function, g(x)? State its y-intercept, domain, and range. 4. The graph of f(x)=(11)−x is reflected about the y-axis and compressed vertically by a factor of 19. What is the equation of the new function, g(x)? State its y-intercept, domain, and range. 5. The graph of f(x)=13x is reflected about the x-axis and shifted upward 6 units. What is the equation of the new function, g(x)? State its y-intercept, domain, and range. 6. The graph of f(x)=(2.69)x is shifted right 2 units, stretched vertically by a factor of 4, reflected about the x-axis, and then shifted downward 4 units. What is the equation of the new function, g(x)? State its y-intercept (to the nearest thousandth), domain, and range. 7. The graph of f(x)=−10(12)x−1+3 is shifted downward 3 units, and then shifted left 1 units, stretched vertically by a factor of 5, and reflected about the x-axis. What is the equation of the new function, g(x)? State its y-intercept, domain, and range.

Let's work through these transformations step by step:

3. Reflection about the y-axis and vertical stretch by a factor of 3:

Original Function: $�(�)=3�$

• Reflection about the y-axis: Reflecting about the y-axis changes the sign of the coefficient of $�$, so we have $�(�)=-3�$.

• Vertical stretch by a factor of 3: To vertically stretch the graph by a factor of 3, we multiply the function by 3, giving us $�(�)=-9�$.

Equation of the New Function: $�(�)=-9�$

Y-Intercept: To find the y-intercept, set $�=0$: $�(0)=-9(0)=0$. The y-intercept is at the point (0, 0).

Domain: The domain of this function is all real numbers because there are no restrictions on $�$.

Range: The range is also all real numbers because the graph extends infinitely in both the positive and negative directions.

4. Reflection about the y-axis and vertical compression by a factor of 19:

Original Function: $�(�)=11-�$

• Reflection about the y-axis: Reflecting about the y-axis changes the sign of the coefficient of $�$, so we have $�(�)=11+�$.

• Vertical compression by a factor of 19: To vertically compress the graph by a factor of 19, we multiply the function by $\frac{1}{19}$, giving us $�(�)=11+\frac{�}{19}$.

Equation of the New Function: $�(�)=11+\frac{�}{19}$

Y-Intercept: To find the y-intercept, set $�=0$: $�(0)=11+\frac{0}{19}=11$. The y-intercept is at the point (0, 11).

Domain: The domain of this function is all real numbers because there are no restrictions on $�$.

Range: The range is all real numbers because the graph extends infinitely in both the positive and negative directions.

5. Reflection about the x-axis and upward shift by 6 units:

Original Function: $�(�)=13�$

• Reflection about the x-axis: Reflecting about the x-axis changes the sign of the function, so we have $�(�)=-13�$.

• Upward shift by 6 units: To shift the graph upward by 6 units, we add 6 to the function, giving us $�(�)=-13�+6$.

Equation of the New Function: $�(�)=-13�+6$

Y-Intercept: To find the y-intercept, set $�=0$: $�(0)=-13(0)+6=6$. The y-intercept is at the point (0, 6).

Domain: The domain of this function is all real numbers because there are no restrictions on $�$.

Range: The range is all real numbers because the graph extends infinitely in both the positive and negative directions.

6. Right shift by 2 units, vertical stretch by a factor of 4, reflection about the x-axis, and downward shift by 4 units:

Original Function: $�(�)=(2.69{)}^{�}$

• Right shift by 2 units: To shift the graph right by 2 units, we replace $�$ with $(�-2)$, giving us $�(�)=(2.69{)}^{�-2}$.

• Vertical stretch by a factor of 4: To vertically stretch the graph by a factor of 4, we multiply the function by 4, giving us $�(�)=4\cdot (2.69{)}^{�-2}$.

• Reflection about the x-axis: Reflecting about the x-axis changes the sign of the function, so we have $�(�)=-4\cdot (2.69{)}^{�-2}$.

• Downward shift by 4 units: To shift the graph downward by 4 units, we subtract 4 from the function, giving us $�(�)=-4\cdot (2.69{)}^{�-2}-4$.

Equation of the New Function: $�(�)=-4\cdot (2.69{)}^{�-2}-4$

Y-Intercept: To find the y-intercept, set $�=0$: $�(0)=-4\cdot (2.69{)}^{-2}-4\approx -4.027$. The y-intercept is approximately (-4.027, 0).

Domain: The domain of this function is all real numbers because there are no restrictions on $�$.

Range: The range is all real numbers because the graph extends infinitely in both the positive and negative directions.

7. Downward shift by 3 units, left shift by 1 unit, vertical stretch by a factor of 5, and reflection about the x-axis:

Original Function: $�(�)=-10\cdot 12^{�-1}+3$

• Downward shift by 3 units: To shift the graph downward by 3 units, we subtract 3 from the function, giving us $�(�)=-10\cdot 12^{�-1}-3$.

• Left shift by 1 unit: To shift the graph to the left by 1 unit, we replace $�$ with $(�+1)$, giving us $�(�)=-10\cdot 12^{�}-3$.

• Vertical stretch by a factor of 5: To vertically stretch the graph by a factor of 5, we multiply the function by 5, giving us $�(�)=-50\cdot 12^{�}-3$.

• Reflection about the x-axis: Reflecting about the x-axis changes the sign of the function, so we have $�(�)=50\cdot 12^{�}-3$.

Equation of the New Function: (g(x) = 50 \