MTH120 College Algebra Chapter 8.3

 8.3 The Parabola

Parabolas are a class of conic sections, and they are one of the most common shapes in mathematics and physics. Parabolas have several important properties and applications in science and engineering.

Graphing parabolas with vertices at the origin is a straightforward process, and these parabolas are typically in the form $�=�{�}^2$ or $�=�{�}^2$, depending on whether they open vertically or horizontally. Here, I'll provide examples and step-by-step instructions for both cases.

Vertical Parabola (Opening Up or Down):

Example 1: Graph the parabola $�=2{�}^2$.

1. Identify the axis of symmetry: In this case, it's the y-axis since the vertex is at the origin.

2. Determine the direction of opening: The coefficient of ${�}^2$ is positive, so the parabola opens upward.

3. Find the vertex: The vertex is at the origin (0,0).

4. Determine additional points: To graph the parabola, you can select additional points by plugging in different x-values. For example, if you plug in x = 1, you get $�=2(1^2)=2$, so you have the point (1, 2).

5. Plot the points: Plot the vertex (0,0) and the additional point (1,2).

6. Draw the parabola: Connect these points with a smooth curve. The parabola opens upward.

Horizontal Parabola (Opening Right or Left):

Example 2: Graph the parabola $�=3{�}^2$.

1. Identify the axis of symmetry: In this case, it's the x-axis since the vertex is at the origin.

2. Determine the direction of opening: The coefficient of ${�}^2$ is positive, so the parabola opens to the right.

3. Find the vertex: The vertex is at the origin (0,0).

4. Determine additional points: To graph the parabola, you can select additional points by plugging in different y-values. For example, if you plug in y = 1, you get $�=3(1^2)=3$, so you have the point (3, 1).

5. Plot the points: Plot the vertex (0,0) and the additional point (3,1).

6. Draw the parabola: Connect these points with a smooth curve. The parabola opens to the right.

These are the basic steps for graphing parabolas with vertices at the origin. Depending on the coefficients in the equations, you can determine the direction of opening (up, down, right, or left) and sketch the corresponding parabola accordingly.

Parabolas with vertices at the origin (0, 0) have standard forms that depend on their orientation, whether they open vertically or horizontally. Here are the standard forms for both cases:

1. Vertical Parabola (Opens Upward or Downward):

The standard form for a vertical parabola with its vertex at the origin is:

• Opens upward: $�=�{�}^2$, where 'a' is a constant that determines the steepness of the parabola. If 'a' is positive, the parabola opens upward. If 'a' is negative, it opens downward.

Example: $�=2{�}^2$ is a vertical parabola that opens upward.

• Opens downward: $�=-�{�}^2$, where 'a' is a positive constant. In this case, 'a' controls the steepness, and the parabola opens downward.

Example: $�=-3{�}^2$ is a vertical parabola that opens downward.

2. Horizontal Parabola (Opens Rightward or Leftward):

The standard form for a horizontal parabola with its vertex at the origin is:

• Opens rightward: $�=�{�}^2$, where 'a' is a positive constant. If 'a' is positive, the parabola opens rightward.

Example: $�=4{�}^2$ is a horizontal parabola that opens rightward.

• Opens leftward: $�=-�{�}^2$, where 'a' is a positive constant. If 'a' is positive, the parabola opens leftward.

Example: $�=-2{�}^2$ is a horizontal parabola that opens leftward.

In these standard forms, 'a' determines the shape and direction of the parabola. A larger absolute value of 'a' makes the parabola steeper, while the sign of 'a' determines whether it opens in the specified direction.

These standard forms are useful for graphing and analyzing parabolas with vertices at the origin. They are versatile and can be applied to various real-world situations and mathematical problems.

Let's look at examples of how to sketch the graph of parabolas centered at the origin (0,0) using standard form equations.

Example 1: Vertical Parabola Opening Upward

Consider the equation $�=4{�}^2$. This is a vertical parabola centered at the origin and opens upward.

1. Identify the vertex: The vertex is at the origin, (0,0).

2. Determine additional points: You can choose different x-values and calculate corresponding y-values to find additional points. For example, if you use x = 1, you get $�=4(1^2)=4$, so you have the point (1, 4).

3. Plot the points: Plot the vertex (0,0) and the additional point (1,4).

4. Draw the parabola: Connect these points with a smooth curve. The parabola opens upward.

Example 2: Vertical Parabola Opening Downward

Now, consider the equation $�=-3{�}^2$. This is a vertical parabola centered at the origin and opens downward.

1. Identify the vertex: The vertex is at the origin, (0,0).

2. Determine additional points: Choose different x-values and calculate corresponding y-values. For example, if you use x = 2, you get $�=-3(2^2)=-12$, so you have the point (2, -12).

3. Plot the points: Plot the vertex (0,0) and the additional point (2, -12).

4. Draw the parabola: Connect these points with a smooth curve. The parabola opens downward.

Example 3: Horizontal Parabola Opening Rightward

Consider the equation $�=3{�}^2$. This is a horizontal parabola centered at the origin and opens rightward.

1. Identify the vertex: The vertex is at the origin, (0,0).

2. Determine additional points: Choose different y-values and calculate corresponding x-values. For example, if you use y = 1, you get $�=3(1^2)=3$, so you have the point (3, 1).

3. Plot the points: Plot the vertex (0,0) and the additional point (3, 1).

4. Draw the parabola: Connect these points with a smooth curve. The parabola opens rightward.

Example 4: Horizontal Parabola Opening Leftward

Now, consider the equation $�=-2{�}^2$. This is a horizontal parabola centered at the origin and opens leftward.

1. Identify the vertex: The vertex is at the origin, (0,0).

2. Determine additional points: Choose different y-values and calculate corresponding x-values. For example, if you use y = 2, you get $�=-2(2^2)=-8$, so you have the point (-8, 2).

3. Plot the points: Plot the vertex (0,0) and the additional point (-8, 2).

4. Draw the parabola: Connect these points with a smooth curve. The parabola opens leftward.

These examples illustrate how to sketch the graph of parabolas with vertices at the origin using their standard form equations. The direction of opening and the shape of the parabola are determined by the coefficients in the equations.

Writing the equation of a parabola in standard form depends on its orientation (whether it's vertical or horizontal) and the position of its vertex. Here are the standard forms for both types of parabolas and examples of how to write their equations:

1. Vertical Parabola (Vertex Form):

The standard form for a vertical parabola with its vertex at (h, k) is:

$(�-ℎ{)}^2=4�(�-�)$

• If the parabola opens upward, 'a' is positive.
• If the parabola opens downward, 'a' is negative.

Example 1: Writing the Equation of an Upward-Opening Vertical Parabola with Vertex (3, 2)

You have a parabola that opens upward with its vertex at (3, 2).

• Since the vertex is (h, k) = (3, 2), you have h = 3 and k = 2.
• Choose a value for 'a'. Let's say 'a' = 2.

Now, plug these values into the standard form:

$(�-3{)}^2=4(2)(�-2)$

Simplify as needed. The equation in standard form is:

$(�-3{)}^2=8(�-2)$

2. Horizontal Parabola (Vertex Form):

The standard form for a horizontal parabola with its vertex at (h, k) is:

$(�-�{)}^2=4�(�-ℎ)$

• If the parabola opens rightward, 'a' is positive.
• If the parabola opens leftward, 'a' is negative.

Example 2: Writing the Equation of a Rightward-Opening Horizontal Parabola with Vertex (-1, 4)

You have a parabola that opens rightward with its vertex at (-1, 4).

• Since the vertex is (h, k) = (-1, 4), you have h = -1 and k = 4.
• Choose a value for 'a'. Let's say 'a' = 3.

Now, plug these values into the standard form:

$(�-4{)}^2=4(3)(�-(-1))$

Simplify as needed. The equation in standard form is:

$(�-4{)}^2=12(�+1)$

These examples illustrate how to write the equation of a parabola in standard form given the vertex and orientation. The values of 'h', 'k', and 'a' determine the specific equation of the parabola.

Graphing parabolas with vertices that are not at the origin involves a few additional steps compared to parabolas with vertices at the origin. To graph these parabolas, you'll need to consider the location of the vertex (h, k), the direction of the opening, and the shape of the parabola. Here are the steps with examples for both vertical and horizontal parabolas:

Example 1: Vertical Parabola with Vertex (h, k)

Suppose you want to graph the parabola with the equation $�=2(�-3{)}^2+1$ with a vertex at (3, 1).

1. Identify the vertex: The vertex is at (3, 1), so $ℎ=3$ and $�=1$.

2. Determine the direction of opening: The coefficient of $(�-ℎ{)}^2$ is positive, which means the parabola opens upward.

3. Additional points: To graph the parabola, choose other x-values and calculate the corresponding y-values. For example, if you use x = 4, you get $�=2(4-3{)}^2+1=3$. So, you have the point (4, 3).

4. Plot the points: Plot the vertex (3, 1) and the additional point (4, 3).

5. Draw the parabola: Connect these points with a smooth curve. The parabola opens upward.

Example 2: Horizontal Parabola with Vertex (h, k)

Suppose you want to graph the parabola with the equation $�=-3(�+2{)}^2+1$ with a vertex at (1, -2).

1. Identify the vertex: The vertex is at (1, -2), so $ℎ=1$ and $�=-2$.

2. Determine the direction of opening: The coefficient of $(�+�{)}^2$ is negative, which means the parabola opens leftward.

3. Additional points: Choose other y-values and calculate the corresponding x-values. For example, if you use y = -1, you get $�=-3(-1+2{)}^2+1=4$. So, you have the point (4, -1).

4. Plot the points: Plot the vertex (1, -2) and the additional point (4, -1).

5. Draw the parabola: Connect these points with a smooth curve. The parabola opens leftward.

These examples illustrate how to graph parabolas with vertices not at the origin. The key is to identify the vertex, determine the direction of opening, select additional points, and then connect those points to create the parabola.

To sketch the graph of a parabola given a standard form equation with a vertex at (h, k), you can follow these steps:

For a Vertical Parabola (opens upward or downward): The standard form for a vertical parabola is of the form $(�-ℎ{)}^2=4�(�-�)$, where (h, k) is the vertex.

1. Identify the vertex: The vertex of the parabola is given by (h, k).

2. Determine the direction of opening: Look at the coefficient 'a' in the equation. If 'a' is positive, the parabola opens upward. If 'a' is negative, the parabola opens downward.

3. Additional points: To sketch the parabola, choose a few more points along the parabolic path. You can select different x-values and calculate the corresponding y-values using the equation.

4. Plot the points: Plot the vertex and the additional points on the graph.

5. Draw the parabola: Connect the points with a smooth curve that follows the shape of the parabola.

For a Horizontal Parabola (opens rightward or leftward): The standard form for a horizontal parabola is of the form $(�-�{)}^2=4�(�-ℎ)$, where (h, k) is the vertex.

1. Identify the vertex: The vertex of the parabola is given by (h, k).

2. Determine the direction of opening: Look at the coefficient 'a' in the equation. If 'a' is positive, the parabola opens rightward. If 'a' is negative, the parabola opens leftward.

3. Additional points: To sketch the parabola, choose a few more points along the parabolic path. You can select different y-values and calculate the corresponding x-values using the equation.

4. Plot the points: Plot the vertex and the additional points on the graph.

5. Draw the parabola: Connect the points with a smooth curve that follows the shape of the parabola.

Here's an example for a vertical parabola:

Example: Sketch the graph of the parabola given by the equation $(�-2{)}^2=4(�+1)$ with a vertex at (2, -1).

1. Vertex: The vertex is (2, -1).

2. Direction of Opening: The coefficient 4 is positive, so the parabola opens upward.

3. Additional Points: Choose a few x-values and calculate the corresponding y-values. For example, if you use x = 3, you get $�=4(3-2{)}^2-1=3$, so you have the point (3, 3).

4. Plot the Points: Plot the vertex (2, -1) and the additional point (3, 3).

5. Draw the Parabola: Connect the points with a smooth curve that follows the shape of the upward-opening parabola.

Follow similar steps for a horizontal parabola, adjusting for the direction of opening and the corresponding points.

Solving applied problems involving parabolas is a common mathematical task in various fields, such as physics, engineering, and economics. These problems often require you to use the properties of parabolas and their standard equations to model real-world situations. Let's go through an example to illustrate this:

Example: Projectile Motion

Suppose you are solving a problem related to projectile motion, such as the motion of a ball being thrown into the air. You want to determine the maximum height reached by the ball, the time it takes to reach that height, and the total time it's in the air.

1. Problem Statement: A ball is thrown upward with an initial velocity of 20 m/s from a height of 1.5 meters above the ground. Determine the maximum height reached by the ball, the time it takes to reach that height, and the total time it's in the air.

2. Understand the Situation: In this scenario, the path of the ball follows a parabolic trajectory. The vertical motion can be modeled by a parabolic equation.

3. Vertical Motion Equation: The vertical motion of the ball is governed by the equation: $�=ℎ+{�}_0�-\frac{1}{2}�{�}^2$ where:

   • $�$ is the vertical position (height above the ground).
   • $ℎ$ is the initial height (1.5 meters).
   • ${�}_0$ is the initial velocity (20 m/s).
   • $�$ is the acceleration due to gravity (approximately 9.81 m/s²).
   • $�$ is the time.

4. Determine the Maximum Height: To find the maximum height, we need to determine when the ball reaches its highest point, which occurs when the vertical velocity (${�}_{�}$) becomes zero. Set ${�}_{�}=0$ and solve for $�$: $0={�}_0-��$ $�=\frac{{�}_0}{�}$ Plug this time into the vertical motion equation to find the maximum height (${�}_{\text{max}}$): ${�}_{\text{max}}=ℎ+{�}_0\left(\frac{{�}_0}{�}\right)-\frac{1}{2}�{\left(\frac{{�}_0}{�}\right)}^2$ Calculate ${�}_{\text{max}}$ to find the answer.

5. Determine the Time to Reach the Maximum Height: The time to reach the maximum height is the same as the time found in the previous step, $�=\frac{{�}_0}{�}$.

6. Determine the Total Time in the Air: The total time in the air is the time it takes for the ball to reach the maximum height and then return to the ground. Since the vertical motion is symmetric, you can simply multiply the time to reach the maximum height by 2.

7. Calculate the Answer: Substitute the known values ($ℎ$, ${�}_0$, and $�$) into the equations and calculate the answers for maximum height, time to reach the maximum height, and total time in the air.

In this example, you apply the properties of parabolic motion to solve a real-world problem involving a projectile. The parabolic equation for vertical motion is used to model the behavior of the ball in the air. Similar approaches can be used to solve other applied problems involving parabolas in various contexts.