MTH120 College Algebra Chapter 5.1

Polynomial and rational functions are two important types of functions in mathematics. Let's explore what each of these functions is and some key characteristics of each:

Polynomial Functions:

A polynomial function is a function that can be represented by a polynomial expression. A polynomial expression is a sum of terms, where each term consists of a constant coefficient multiplied by a variable raised to a non-negative integer exponent. The general form of a polynomial function is:

f(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_2 * x^2 + a_1 * x + a_0

• f(x) represents the polynomial function.
• a_n, a_(n-1), ..., a_2, a_1, a_0 are the coefficients of the polynomial.
• x is the variable.
• n is a non-negative integer, which represents the degree of the polynomial. The highest power of x in the polynomial determines the degree.

Key characteristics of polynomial functions:

• Polynomial functions are continuous over their entire domain.
• The degree of a polynomial function tells you the highest number of turning points it can have.
• The behavior of the polynomial at the far ends of the x-axis (as x approaches positive or negative infinity) depends on the leading term of the polynomial.

Examples of polynomial functions:

• Linear function: f(x) = ax + b (degree 1)
• Quadratic function: f(x) = ax^2 + bx + c (degree 2)
• Cubic function: f(x) = ax^3 + bx^2 + cx + d (degree 3)
• Higher-degree polynomials: f(x) = ... (degree greater than 3)

Rational Functions:

A rational function is a function that is defined as the ratio of two polynomial functions. The general form of a rational function is:

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f(x) = P(x) / Q(x)

• f(x) represents the rational function.
• P(x) and Q(x) are polynomial functions.
• The domain of the rational function is all real numbers except the values of x that make the denominator, Q(x), equal to zero (these are called "excluded values" or "vertical asymptotes").

Key characteristics of rational functions:

• Rational functions may have vertical asymptotes at the excluded values of x where the denominator equals zero.
• Horizontal asymptotes can exist for rational functions, depending on the degree of the numerator and denominator.
• Rational functions can have holes (removable discontinuities) in their graphs.

Examples of rational functions:

• Simple rational function: f(x) = (x + 1) / (x - 2)
• Rational function with vertical asymptotes: f(x) = 1 / (x - 3)
• Rational function with horizontal asymptotes: f(x) = (3x^2 + 2) / (x^2 + 1)
• Rational function with a hole: f(x) = (x^2 - 4) / (x - 2)

Both polynomial and rational functions are essential in various branches of mathematics and science. They are used to model real-world phenomena, solve equations, and analyze data. Understanding their properties and behavior is fundamental to mathematical problem-solving and modeling.

5.1 Quadratic Functions:

Parabolas are a fundamental type of curve in mathematics, and they are commonly encountered in various applications. To recognize the characteristics of parabolas, it's important to understand their general form and key properties. Here are some key characteristics of parabolas:

1. General Form of a Parabola:

• The general form of a parabola is given by the equation:

  
  y = ax^2 + bx + c
  

• In this equation:
  • y is the dependent variable.
  • x is the independent variable.
  • a, b, and c are constants, with a ≠ 0.
  • The term ax^2 represents the quadratic term, which is essential for the parabolic shape.

2. Axis of Symmetry:

• A parabola has an axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves.
• The equation of the axis of symmetry is x = -b/(2a).

3. Vertex:

• The vertex of a parabola is the point where the parabola reaches its minimum (if a > 0) or maximum (if a < 0) value.
• The coordinates of the vertex are given by (h, k), where h = -b/(2a) and k is the value of the function at the vertex.

4. Direction of Opening:

• The direction in which the parabola opens depends on the sign of the coefficient a:
  • If a is positive (a > 0), the parabola opens upward, and the vertex is the minimum point.
  • If a is negative (a < 0), the parabola opens downward, and the vertex is the maximum point.

5. Axis of Symmetry as a Line of Reflection:

• The axis of symmetry serves as a line of reflection. Any point (x, y) on one side of the parabola is reflected across the axis of symmetry to a corresponding point on the other side of the parabola.

6. Focus and Directrix (for Conic Sections):

• In the context of conic sections, parabolas have a focus and a directrix.
• The focus is a fixed point, and the directrix is a fixed line. The parabola is the set of points such that the distance from each point to the focus is equal to the perpendicular distance from that point to the directrix.

7. The Vertex Form of a Parabola:

• The vertex form of a parabola is given by:

  
  y = a(x - h)^2 + k
  

• In this form, (h, k) is the vertex of the parabola.

8. The Standard Form of a Parabola:

• The standard form of a parabola is given by:

  
  (x - h)^2 = 4a(y - k)
  

• In this form, (h, k) is the vertex of the parabola, and 4a represents the focal width (distance from the vertex to the focus).

These characteristics are essential for recognizing and understanding the properties of parabolas. Parabolas are commonly seen in algebra, geometry, calculus, physics, engineering, and other fields, making them a fundamental concept in mathematics and science.

Understanding how the graphs of parabolas are related to their quadratic functions is essential in algebra and calculus. The relationship between the two helps you visualize and analyze quadratic equations and their solutions. Here's how the graphs of parabolas are related to their quadratic functions:

1. Vertex and Axis of Symmetry:

• The vertex of a parabola is a key feature that corresponds to important information in the quadratic function.
• In the quadratic function f(x) = ax^2 + bx + c, the vertex of the parabola is given by (h, k), where h and k are the coordinates of the vertex.
• The x-coordinate of the vertex, h, is determined by the equation h = -b/(2a). This is also the equation for the axis of symmetry.
• The y-coordinate of the vertex, k, is the value of the quadratic function at the vertex, i.e., k = f(h).

2. Direction of Opening:

• The direction in which the parabola opens is determined by the sign of the leading coefficient a in the quadratic function.
• If a is positive (a > 0), the parabola opens upward, and the vertex is the minimum point.
• If a is negative (a < 0), the parabola opens downward, and the vertex is the maximum point.

3. Axis of Symmetry as a Line of Reflection:

• The axis of symmetry, given by x = h, is a vertical line that divides the parabola into two symmetrical halves.
• Any point (x, y) on one side of the parabola is reflected across the axis of symmetry to a corresponding point on the other side of the parabola.

4. Roots or Solutions:

• The roots or solutions of a quadratic equation correspond to the x-values where the parabola intersects the x-axis.
• These roots can be found by solving the quadratic equation ax^2 + bx + c = 0.
• The number of real roots and their nature (real, equal, or complex) can be determined by the discriminant b^2 - 4ac.

5. Characteristics of the Parabola:

• The shape, orientation, and location of the parabola on the coordinate plane are determined by the coefficients a, b, and c in the quadratic function.
• For example, the value of a determines how "narrow" or "wide" the parabola is, while c determines its vertical shift.

6. Focus and Directrix (for Conic Sections):

• In the context of conic sections, parabolas have a focus and a directrix that are related to the quadratic function.
• The focus and directrix help define the geometric properties of the parabola.

Understanding these relationships between the graphs of parabolas and their quadratic functions allows you to analyze and interpret the behavior of quadratic equations more effectively. It also provides insight into how to manipulate quadratic functions to achieve specific graph characteristics, such as shifting, stretching, or reflecting the parabola.

Here are some examples of quadratic functions along with their corresponding parabolic graphs:

1. Quadratic Function with Vertex at the Origin:

• Quadratic Function: f(x) = x^2
• Graph: This is a simple upward-opening parabola with its vertex at the origin (0, 0).

2. Quadratic Function with Vertex at (2, 3):

• Quadratic Function: f(x) = (x - 2)^2 + 3
• Graph: This is an upward-opening parabola with its vertex at (2, 3).

3. Quadratic Function with Vertex at (-1, -4):

• Quadratic Function: f(x) = (x + 1)^2 - 4
• Graph: This is an upward-opening parabola with its vertex at (-1, -4).

4. Quadratic Function with Vertex at (3, -2) and Horizontal Stretch:

• Quadratic Function: f(x) = 2(x - 3)^2 - 2
• Graph: This is an upward-opening parabola with its vertex at (3, -2). The coefficient of 2 in front of the (x - 3)^2 term stretches the parabola horizontally.

5. Quadratic Function with Vertex at (1, 4) and Downward Opening:

• Quadratic Function: f(x) = -2(x - 1)^2 + 4
• Graph: This is a downward-opening parabola with its vertex at (1, 4).

These examples demonstrate the relationship between the quadratic functions and their parabolic graphs. The vertex, direction of opening, and coefficients in the quadratic function determine the key characteristics of the parabola, including its shape, orientation, and position on the coordinate plane.

Quadratic functions can be represented in various forms, each offering different insights into the function's properties and characteristics. The three most common forms of quadratic functions are the standard form, vertex form, and factored form. Here's an explanation of each form:

1. Standard Form:

The standard form of a quadratic function is given as:

f(x) = ax^2 + bx + c

• In this form:
  • a, b, and c are constants, where a cannot be zero (since a quadratic function must have a leading term with x^2).
  • The coefficient a determines the direction of opening (positive a for upward, negative a for downward), and it also affects the width of the parabola.
  • The coefficients b and c determine the horizontal and vertical shifts of the parabola, respectively.

2. Vertex Form:

The vertex form of a quadratic function is given as:

f(x) = a(x - h)^2 + k

• In this form:
  • a, h, and k are constants.
  • (h, k) represents the coordinates of the vertex of the parabola.
  • The value of a determines the direction of opening and the steepness of the parabola.

This form explicitly shows the vertex of the parabola and can make it easier to identify the vertex and the direction of opening. It is particularly useful for graphing and transformations.

3. Factored Form:

The factored form of a quadratic function is given as:

f(x) = a(x - r1)(x - r2)

• In this form:
  • a is a constant.
  • r1 and r2 are the roots or solutions of the quadratic equation ax^2 + bx + c = 0. They represent the x-intercepts of the parabola.

The factored form directly reveals the roots of the quadratic equation and can be useful for finding the x-intercepts of the parabola or solving quadratic equations.

Each of these forms provides a different perspective on the quadratic function and its behavior. Converting between these forms can be useful for various purposes, such as graphing, finding the vertex, or solving quadratic equations. The choice of form depends on the specific problem and what information you want to extract from the quadratic function.

To find the vertex of a quadratic function in general form (i.e., the form f(x) = ax^2 + bx + c), you can use the following formula for the x-coordinate of the vertex:

h = -b / (2a)

Once you have calculated h, you can find the y-coordinate of the vertex by plugging h into the quadratic function:

k = f(h) = a(h)^2 + b(h) + c

Here are the steps to find the vertex of the parabola:

1. Identify the coefficients a, b, and c from the given quadratic function f(x) = ax^2 + bx + c.

2. Calculate h using the formula h = -b / (2a).

3. Calculate k by plugging the value of h into the quadratic function:

   
   k = a(h)^2 + b(h) + c
   

4. The vertex of the parabola is (h, k).

Let's work through an example:

Example: Given the quadratic function f(x) = 2x^2 - 8x + 6, find the vertex of the parabola.

1. Identify the coefficients:

   • a = 2
   • b = -8
   • c = 6

2. Calculate h using h = -b / (2a):

   
   h = -(-8) / (2 * 2) = 8 / 4 = 2
   

3. Calculate k by plugging h into the quadratic function:

   
   k = 2(2)^2 - 8(2) + 6 = 8 - 16 + 6 = -2
   

4. The vertex of the parabola is (2, -2).

So, the vertex of the parabola defined by the quadratic function f(x) = 2x^2 - 8x + 6 is (2, -2).

To find the domain and range of a quadratic function, you need to consider the characteristics of quadratic functions and the properties of their graphs. Here's how to find the domain and range of a quadratic function:

1. Domain of a Quadratic Function:

The domain of a quadratic function is the set of all real numbers (all possible values of x) unless there are specific restrictions due to the context of the problem. In other words, the domain is usually the entire real number line unless there are excluded values that make the function undefined.

Quadratic functions, in their standard form f(x) = ax^2 + bx + c, are defined for all real values of x. Therefore, the domain is:

Domain: All real numbers (-∞, ∞)

There are no excluded values unless stated otherwise by the problem.

2. Range of a Quadratic Function:

The range of a quadratic function depends on the direction of opening (upward or downward) and the vertex of the parabola.

• If the quadratic function opens upward (meaning a is positive), the range is all real numbers greater than or equal to the y-coordinate of the vertex (the minimum point).
• If the quadratic function opens downward (meaning a is negative), the range is all real numbers less than or equal to the y-coordinate of the vertex (the maximum point).

To find the y-coordinate of the vertex (k) and determine the range:

1. Convert the quadratic function into vertex form f(x) = a(x - h)^2 + k, where (h, k) is the vertex.

2. Identify the sign of the coefficient a:

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

3. The range is determined as follows:

   • If the parabola opens upward (a is positive), the range is [k, ∞).
   • If the parabola opens downward (a is negative), the range is (-∞, k].

Example: Given the quadratic function f(x) = 2x^2 - 8x + 6, find its domain and range.

1. The domain is all real numbers (-∞, ∞) because there are no excluded values.

2. To find the range, first convert the function to vertex form:

   
   f(x) = 2(x^2 - 4x) + 6 = 2(x^2 - 4x + 4) + 6 - 2(4) = 2(x - 2)^2 + 2
   

3. The coefficient of x^2 is positive (a = 2), indicating that the parabola opens upward. The vertex is (h, k) = (2, 2).

4. The range is [2, ∞) because the parabola opens upward.

So, for the quadratic function f(x) = 2x^2 - 8x + 6, the domain is all real numbers, and the range is [2, ∞).




To find the domain and range of a quadratic function, you can follow these steps:

1. Domain of a Quadratic Function:

The domain of a quadratic function is the set of all real numbers unless there are specific restrictions due to the context of the problem. In other words, the domain is typically the entire real number line unless there are excluded values that make the function undefined.

Quadratic functions, in their standard form f(x) = ax^2 + bx + c, are defined for all real values of x. Therefore, the domain is:

Domain: All real numbers (-∞, ∞)

There are no excluded values unless stated otherwise by the problem.

2. Range of a Quadratic Function:

The range of a quadratic function depends on the direction of the parabolic curve's opening (upward or downward) and the location of the vertex.

• If the quadratic function opens upward (meaning a is positive), the range includes all real numbers greater than or equal to the y-coordinate of the vertex (the minimum point).
• If the quadratic function opens downward (meaning a is negative), the range includes all real numbers less than or equal to the y-coordinate of the vertex (the maximum point).

To find the range:

1. Convert the quadratic function into vertex form f(x) = a(x - h)^2 + k, where (h, k) is the vertex.

2. Identify the sign of the coefficient a:

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

3. The range is determined as follows:

   • If the parabola opens upward (a is positive), the range is [k, ∞).
   • If the parabola opens downward (a is negative), the range is (-∞, k].

Example:

Given the quadratic function f(x) = -2x^2 + 4x + 6, find its domain and range.

1. The domain is all real numbers (-∞, ∞) because there are no excluded values.

2. To find the range, first convert the function to vertex form:

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   f(x) = -2(x^2 - 2x) + 6 = -2(x^2 - 2x + 1) + 6 + 2(1) = -2(x - 1)^2 + 8
   

3. The coefficient of x^2 is negative (a = -2), indicating that the parabola opens downward. The vertex is (h, k) = (1, 8).

4. The range is (-∞, 8] because the parabola opens downward.

So, for the quadratic function f(x) = -2x^2 + 4x + 6, the domain is all real numbers, and the range is (-∞, 8].




To determine the maximum or minimum values of a quadratic function, you need to consider the direction of the parabolic curve (whether it opens upward or downward) and locate its vertex. The vertex represents either the minimum or maximum point of the quadratic function, depending on the direction of the curve.

Here are the steps to determine the maximum or minimum values of a quadratic function:

1. Identify the Direction of the Parabolic Curve:

• Look at the coefficient of the leading term (a) in the quadratic function f(x) = ax^2 + bx + c:
  • If a is positive (a > 0), the parabola opens upward, and there is a minimum value.
  • If a is negative (a < 0), the parabola opens downward, and there is a maximum value.

2. Find the Vertex of the Parabola:

• The vertex (h, k) of the parabola can be found using the formula:

  
  h = -b / (2a)
  k = f(h) = a(h)^2 + b(h) + c
  

  where h and k represent the x- and y-coordinates of the vertex, respectively.

3. Determine the Maximum or Minimum Value:

• Depending on the direction of the parabolic curve:
  • If the parabola opens upward (a > 0), the minimum value of the function is k, which is the y-coordinate of the vertex. The minimum value occurs at the vertex.
  • If the parabola opens downward (a < 0), the maximum value of the function is k, which is the y-coordinate of the vertex. The maximum value occurs at the vertex.

Example 1 (Finding the Minimum Value):

Given the quadratic function f(x) = x^2 - 4x + 3, find its minimum value.

1. The coefficient of the leading term is positive (a = 1), so the parabola opens upward.

2. Calculate the vertex using the formulas:

   
   h = -b / (2a) = -(-4) / (2 * 1) = 2
   k = f(h) = 1(2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1
   

3. The minimum value occurs at the vertex, which is (2, -1).

So, the minimum value of the quadratic function f(x) = x^2 - 4x + 3 is -1, and it occurs at the point (2, -1).

Example 2 (Finding the Maximum Value):

Given the quadratic function f(x) = -x^2 + 4x - 3, find its maximum value.

1. The coefficient of the leading term is negative (a = -1), so the parabola opens downward.

2. Calculate the vertex using the formulas:

   
   h = -b / (2a) = -4 / (-2) = 2
   k = f(h) = -1(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1
   

3. The maximum value occurs at the vertex, which is (2, 1).

So, the maximum value of the quadratic function f(x) = -x^2 + 4x - 3 is 1, and it occurs at the point (2, 1).

Quadratic equations are often used in business and economics to model various real-world scenarios, including revenue optimization. To find the maximum revenue in such applications, you can follow these general steps:

1. Define the Variables:

   • Let x represent the quantity of the product or service sold.
   • Let R(x) represent the revenue generated when x units are sold.
   • Specify the price per unit, which is typically a constant, and denote it as p.

2. Write the Revenue Function:

   • The revenue generated by selling x units at a price p per unit can be expressed as:

     
     R(x) = p * x
     

3. Introduce Cost:

   • If applicable, introduce the cost function C(x) that represents the total cost associated with producing x units. The cost function is typically quadratic and may include fixed and variable costs.

4. Write the Profit Function:

   • The profit is the difference between revenue and cost:

     
     P(x) = R(x) - C(x)
     

5. Find the Maximum:

   • To find the maximum profit (and, indirectly, the maximum revenue), determine the quantity of units x that maximizes the profit function P(x).
   • Set up the profit function as a quadratic equation and find its vertex. The x-coordinate of the vertex corresponds to the quantity that maximizes profit and revenue.

6. Interpret the Result:

   • Once you find the value of x that maximizes profit (and revenue), you can calculate the corresponding maximum revenue by plugging this x value into the revenue function R(x).

Here's a simplified example to illustrate this process:

Example: Revenue Maximization

Suppose a company sells widgets for $20 each, and their monthly cost function is given by C(x) = 0.5x^2 + 100x + 500, where x is the number of widgets produced and C(x) represents the total cost. Find the quantity that maximizes revenue and calculate the maximum revenue.

1. Define the variables:

   • x represents the quantity of widgets sold.
   • R(x) represents the revenue generated when x widgets are sold.
   • p is the price per widget, which is $20 in this case.

2. Write the revenue function:

   
   R(x) = p * x = 20x
   

3. Write the profit function:

   
   P(x) = R(x) - C(x) = 20x - (0.5x^2 + 100x + 500)
        = -0.5x^2 - 80x - 500
   

4. Find the maximum profit:

   • To find the maximum of the profit function, determine the vertex. The x-coordinate of the vertex gives you the quantity that maximizes profit and revenue.
   • The vertex can be found using the formula h = -b / (2a), where in this case, a = -0.5 and b = -80.

   
   h = -(-80) / (2 * (-0.5)) = 80 / 1 = 80
   

   So, x = 80 represents the quantity that maximizes profit (and revenue).

5. Calculate the maximum revenue:

   • Plug the x value into the revenue function R(x):

   
   R(80) = 20 * 80 = $1600
   

6. Interpret the result:

   • The quantity of 80 widgets maximizes revenue, and the maximum revenue achievable is $1600.

In this example, by using a quadratic equation to model revenue and profit, you can determine the quantity of widgets to produce to maximize revenue effectively.

To find the x- and y-intercepts of a quadratic function, you need to determine the points where the graph of the function intersects the x-axis (x-intercepts) and the y-axis (y-intercept). Here's how to find them:

1. X-Intercepts (Roots or Zeros):

X-intercepts are the points where the graph of the quadratic function crosses the x-axis. To find the x-intercepts, set the function equal to zero and solve for x. These are also known as the roots or zeros of the function.

1. Start with the quadratic function in general form: f(x) = ax^2 + bx + c.

2. Set the function equal to zero:

   
   ax^2 + bx + c = 0
   

3. Solve the quadratic equation for x using the quadratic formula or factoring.

   a. Using the quadratic formula:

   
   x = (-b ± √(b² - 4ac)) / (2a)
   

   This formula gives the x-values where the graph intersects the x-axis.

   b. By factoring (if possible): Factor the quadratic equation into two binomials and set each binomial equal to zero to find the x-values.

4. The x-values obtained from the solutions are the x-intercepts.

2. Y-Intercept:

The y-intercept is the point where the graph of the quadratic function crosses the y-axis. To find the y-intercept, set x equal to zero and calculate the function's value (f(0)).

1. Start with the quadratic function in general form: f(x) = ax^2 + bx + c.

2. Substitute x = 0 into the function:

   
   f(0) = a(0)^2 + b(0) + c = c
   

3. The value c is the y-intercept, and the point is (0, c).

So, to summarize:

• X-intercepts: Solve the quadratic equation ax^2 + bx + c = 0 for x.
• Y-intercept: Substitute x = 0 into the function to find f(0).

Let's work through an example:

Example: Find the x- and y-intercepts of the quadratic function f(x) = x^2 - 4x - 5.

1. Start with the quadratic function:

   
   f(x) = x^2 - 4x - 5
   

2. Find the x-intercepts by solving the equation x^2 - 4x - 5 = 0. You can use the quadratic formula:

   
   x = (-(-4) ± √((-4)² - 4(1)(-5))) / (2(1))
   x = (4 ± √(16 + 20)) / 2
   x = (4 ± √36) / 2
   x = (4 ± 6) / 2
   

   The solutions are x = 5 and x = -1, so the x-intercepts are (5, 0) and (-1, 0).

3. Find the y-intercept by substituting x = 0 into the function:

   
   f(0) = (0)^2 - 4(0) - 5 = -5
   

   The y-intercept is -5, so the point is (0, -5).

In this example, the x-intercepts are (5, 0) and (-1, 0), and the y-intercept is (0, -5).

To rewrite a quadratic equation in standard form, you need to express it as:

ax^2 + bx + c = 0

Here's how you can do that:

1. Start with the quadratic equation in any form it is given (vertex form, factored form, or another form). Let's assume you have a quadratic equation in the form f(x) = ax^2 + bx + c.

2. If the equation is not already set to zero (= 0), move all terms to one side of the equation so that it is equal to zero. To do this, subtract the entire equation by the expression on the right side.

3. Ensure that the coefficients a, b, and c are in numerical order. Typically, you arrange them in decreasing order of the exponent of x (i.e., a is the coefficient of x^2, b is the coefficient of x, and c is the constant term).

4. Your quadratic equation is now in standard form: ax^2 + bx + c = 0.

Here's an example to illustrate this process:

Example: Rewrite the quadratic equation f(x) = (x - 2)(x + 3) in standard form.

1. Start with the quadratic equation:

   
   f(x) = (x - 2)(x + 3)
   

2. Move all terms to one side to make it equal to zero:

   
   f(x) - (x - 2)(x + 3) = 0
   

3. Expand the expression (x - 2)(x + 3) and simplify the equation:

   
   f(x) - (x^2 + 3x - 2x - 6) = 0
   f(x) - x^2 + x - 6 = 0
   

4. Ensure that the coefficients are in numerical order:

   
   -x^2 + x + f(x) - 6 = 0
   

Now, the quadratic equation is in standard form: -x^2 + x + f(x) - 6 = 0. If you have a specific value for f(x), you can substitute it in to get a specific quadratic equation in standard form.

Remember that standard form is a way to express quadratic equations so that they are in a common format, which makes it easier to work with and solve them.

To find the x-intercepts of a quadratic function by rewriting it in standard form, you should follow these steps:

1. Start with the quadratic function in its current form. Let's assume you have a quadratic function of the form f(x) = ax^2 + bx + c.

2. Set the function equal to zero (= 0) because you want to find the points where the graph crosses the x-axis:

   
   ax^2 + bx + c = 0
   

3. Rearrange the equation so that the terms are ordered in decreasing powers of x. The standard form for a quadratic equation is ax^2 + bx + c = 0, so ensure that your equation matches this form. To do this, just move all terms to the left side of the equation:

   
   ax^2 + bx + c - 0 = 0
   ax^2 + bx + c = 0
   

4. Now, your quadratic function is in standard form. The coefficients a, b, and c are in numerical order, and it's ready for you to apply the quadratic formula or factor it if possible to find the x-intercepts.

5. You can find the x-intercepts by either:

   • Applying the quadratic formula:

     
     x = (-b ± √(b^2 - 4ac)) / (2a)
     

     This formula will give you the x-values where the graph intersects the x-axis. These are your x-intercepts.

   • Factoring the quadratic equation: Factor the quadratic equation into two binomials and set each binomial equal to zero to find the x-values.

Here's an example to illustrate this process:

Example: Find the x-intercepts of the quadratic function f(x) = 2x^2 - 5x - 3 by rewriting it in standard form.

1. Start with the quadratic function:

   
   f(x) = 2x^2 - 5x - 3
   

2. Set it equal to zero:

   2x^2 - 5x - 3 = 0
   

3. Ensure that the terms are ordered correctly:

   2x^2 - 5x - 3 = 0
   

4. The equation is now in standard form.

5. To find the x-intercepts, you can either use the quadratic formula or factor the equation.

   Using the quadratic formula:

   
   x = (-(-5) ± √((-5)^2 - 4(2)(-3))) / (2(2))
   x = (5 ± √(25 + 24)) / 4
   x = (5 ± √49) / 4
   x = (5 ± 7) / 4
   

   This gives two solutions: x = 3 and x = -1. So, the x-intercepts are (3, 0) and (-1, 0).

So, by rewriting the quadratic function in standard form, you can find the x-intercepts by applying the quadratic formula or factoring the equation.

To find the x-intercepts (also called roots or zeros) of a parabola, you need to determine the values of x where the parabola crosses the x-axis. The x-intercepts are the points where the y-coordinate is zero. Here are the general steps to find the x-intercepts of a parabola:

1. Start with the equation of the parabola. A general quadratic equation is typically written in the form:

   
   y = ax^2 + bx + c
   

   where a, b, and c are constants.

2. Set the equation equal to zero, as you are looking for the values of x where y is zero:

   
   ax^2 + bx + c = 0
   

3. Use the quadratic formula or factor the equation to solve for x. The quadratic formula is as follows:

   
   x = (-b ± √(b² - 4ac)) / (2a)
   

   where a, b, and c are the coefficients from the quadratic equation.

4. Solve for x to find the x-intercepts.

   a. If the discriminant b² - 4ac is positive, you will have two real x-intercepts. b. If the discriminant is zero, you will have one real x-intercept (a repeated root). c. If the discriminant is negative, there are no real x-intercepts (the parabola does not cross the x-axis).

5. Write the x-intercepts as ordered pairs in the form (x, 0).

Here's an example to illustrate this process:

Example: Find the x-intercepts of the parabola defined by the equation y = 3x^2 - 6x - 9.

1. Start with the equation of the parabola:

   
   y = 3x^2 - 6x - 9
   

2. Set it equal to zero:

   3x^2 - 6x - 9 = 0
   

3. Use the quadratic formula to solve for x. In this case, a = 3, b = -6, and c = -9. Apply the quadratic formula:

   
   x = (-(-6) ± √((-6)² - 4(3)(-9))) / (2(3))
   x = (6 ± √(36 + 108)) / 6
   x = (6 ± √144) / 6
   x = (6 ± 12) / 6
   

4. Solve for x:

   • For the plus sign:

     
     x = (6 + 12) / 6 = 18 / 6 = 3
     

   • For the minus sign:

     
     x = (6 - 12) / 6 = -6 / 6 = -1
     

5. Write the x-intercepts as ordered pairs:

   • The x-intercepts are (3, 0) and (-1, 0).

So, the x-intercepts of the parabola defined by y = 3x^2 - 6x - 9 are (3, 0) and (-1, 0). These are the points where the parabola crosses the x-axis.