Graphing Parabolas: Mastering Intercept Form

Algebra 2 Grades High School 7:50 Video
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Lesson Description

Learn how to graph quadratic functions (parabolas) expressed in intercept form. This lesson covers finding x-intercepts, the axis of symmetry, the vertex, domain, range, and intervals of increasing and decreasing.

Video Resource

Graphing Parabolas in Intercept Form (3 Examples)

Mario's Math Tutoring

Duration: 7:50
Watch on YouTube

Key Concepts

  • Intercept Form of a Quadratic Equation: y = a(x - p)(x - q)
  • X-intercepts: (p, 0) and (q, 0)
  • Axis of Symmetry: x = (p + q) / 2
  • Vertex: The point on the axis of symmetry; found by substituting the x-value of the axis of symmetry into the equation.
  • Domain and Range of a Quadratic Function
  • Maximum or Minimum Value: The y-coordinate of the vertex
  • Intervals of Increasing and Decreasing

Learning Objectives

  • Students will be able to identify the x-intercepts of a parabola given its equation in intercept form.
  • Students will be able to determine the axis of symmetry and vertex of a parabola in intercept form.
  • Students will be able to graph a parabola in intercept form.
  • Students will be able to determine the domain and range of a parabola.
  • Students will be able to identify intervals where the function is increasing or decreasing.
  • Students will be able to determine whether the parabola has a maximum or minimum value and identify that value.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the standard form of a quadratic equation and its factored form. Introduce the intercept form y = a(x - p)(x - q) and explain how 'p' and 'q' directly relate to the x-intercepts. Briefly discuss the significance of the 'a' value in determining the direction of the parabola (opening up or down) and its stretch.
  • Finding X-Intercepts (5 mins)
    Explain how to find the x-intercepts by setting each factor (x - p) and (x - q) equal to zero and solving for x. Provide a quick example to illustrate the process.
  • Axis of Symmetry and Vertex (10 mins)
    Demonstrate how to calculate the axis of symmetry using the formula x = (p + q) / 2. Explain that the axis of symmetry is the vertical line that divides the parabola into two symmetrical halves. Show how to find the vertex by substituting the x-value of the axis of symmetry back into the original equation to find the corresponding y-value.
  • Graphing the Parabola (10 mins)
    Walk through the process of plotting the x-intercepts and the vertex on a coordinate plane. Explain how the 'a' value determines whether the parabola opens upwards (a > 0) or downwards (a < 0). If needed, find additional points (like the y-intercept) to improve the accuracy of the sketch. Go through each example in the video, pausing to allow students to work ahead.
  • Domain, Range, Increasing/Decreasing Intervals, and Max/Min (10 mins)
    Explain that the domain of any parabola is all real numbers. Show how to determine the range based on whether the parabola opens up or down and the y-coordinate of the vertex. Discuss how to identify the intervals where the function is increasing and decreasing. Relate the vertex to the maximum or minimum value of the function.
  • Practice Problems (10 mins)
    Provide students with practice problems involving graphing parabolas in intercept form. Have them work individually or in pairs. Circulate the room to provide assistance and answer questions.

Interactive Exercises

  • Graphing Challenge
    Divide students into teams. Provide each team with a quadratic equation in intercept form. Teams race to correctly identify the x-intercepts, axis of symmetry, vertex, and a sketch of the graph. Award points for accuracy and speed.

Discussion Questions

  • How does the 'a' value in the intercept form affect the shape and direction of the parabola?
  • Why is the axis of symmetry important when graphing a parabola?
  • How can you determine the range of a parabola if you know its vertex and direction?
  • Can a parabola have two y-intercepts? Why or why not?

Skills Developed

  • Graphing quadratic functions
  • Identifying key features of parabolas (x-intercepts, vertex, axis of symmetry)
  • Analyzing the domain and range of a function
  • Determining intervals of increasing and decreasing
  • Problem-solving

Multiple Choice Questions

Question 1:

What is the intercept form of a quadratic equation?

Correct Answer: y = a(x - p)(x - q)

Question 2:

The x-intercepts of the parabola y = 2(x - 3)(x + 1) are:

Correct Answer: (3, 0) and (-1, 0)

Question 3:

The axis of symmetry for the parabola y = (x - 2)(x + 4) is:

Correct Answer: x = -1

Question 4:

If a parabola in intercept form has 'a' < 0, it opens:

Correct Answer: Downwards

Question 5:

The vertex of a parabola represents:

Correct Answer: The maximum or minimum point

Question 6:

What is the domain of all parabolas?

Correct Answer: All real numbers

Question 7:

The range of a parabola y = (x-1)(x+3) is

Correct Answer: y ≥ -4

Question 8:

For what values of x is the function y = (x-2)(x-4) increasing?

Correct Answer: x > 3

Question 9:

Given the quadratic equation y = -2(x + 1)(x - 3), does the graph have a maximum or minimum value?

Correct Answer: Maximum

Question 10:

The x-coordinate of the vertex is always found on the:

Correct Answer: axis of symmetry

Fill in the Blank Questions

Question 1:

The intercept form of a quadratic equation is y = a(x - p)(x - q), where p and q are the ________.

Correct Answer: x-intercepts

Question 2:

The line that divides the parabola into two symmetrical halves is called the ________.

Correct Answer: axis of symmetry

Question 3:

If the 'a' value in the intercept form is positive, the parabola opens ________.

Correct Answer: upwards

Question 4:

The highest or lowest point on the parabola is called the ________.

Correct Answer: vertex

Question 5:

The set of all possible y-values of a function is called the ________.

Correct Answer: range

Question 6:

To find the axis of symmetry, you can average the two ________.

Correct Answer: x-intercepts

Question 7:

If the 'a' value is negative, the parabola has a ________ value at its vertex.

Correct Answer: maximum

Question 8:

The set of all possible x-values of a function is called the ________.

Correct Answer: domain

Question 9:

The parabola is ________ on the left side of the axis of symmetry if it opens downwards.

Correct Answer: increasing

Question 10:

In the intercept form y=a(x-p)(x-q), setting (x-p) equal to zero allows us to find one of the _______ of the parabola.

Correct Answer: x-intercepts

Educational Standards

A2.F.1.3:Graph a quadratic function. Identify the domain, range, x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. A2.A.1.1:Use mathematical models to represent quadratic relationships and solve using factoring, completing the square, the quadratic formula, and various methods (including graphing calculator or other appropriate technology). Find non-real roots when they exist. A2.F.1.5:Analyze the graph of a polynomial function by identifying the domain, range, intercepts, zeros, relative maxima, relative minima, and intervals of increase and decrease.

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