Lesson 22

Features of Parabolas

These materials, when encountered before Algebra 1, Unit 7, Lesson 22 support success in that lesson.

22.1: Matching Quadratic Graphs (5 minutes)

Warm-up

In this partner activity, students take turns matching quadratic equations written in factored and standard forms with their graphs. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). In the associated Algebra 1 lesson, students examine the vertex form of quadratic equations, so this will help them practice the other forms they have encountered.

Launch

Arrange students in groups of 2. Ask students to take turns: the first partner identifies a match and explains why they think it is a match, while the other listens and works to understand. When both partners agree on the match, they switch roles.

Student Facing

Match the equation to the graph. Be prepared to explain your reasoning.

  1. \(y = x^2+x\)
  2. \(y = \text{-}x^2 - 3x\)
  3. \(y = (x-1)(x+5)\)
  4. \(y = x^2 + 5x +1\)

    A

    Parabola facing up with vertex in quadrant 3, near -2 comma -9

    B

    Parabola facing up with vertex in quadrant 3, near -2 comma -5

    C

    Parabola facing down with vertex in quadrant 2, near -2 comma 2.

    D

    Parabola facing up with vertex in quadrant 3, just a little to the left and little below origin 

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Once all groups have completed the matching, discuss the following:

  • “Which matches were tricky? Explain why.”
  • “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

22.2: Features of a Quadratic Graph (20 minutes)

Activity

In this activity, students examine key features of the graph of a quadratic, including intercepts, vertex, and line of symmetry. In the associated Algebra 1 lesson, students begin working with quadratic equations in vertex form. Students use appropriate tools strategically (MP5) when they use graphing technology to graph the given functions.

Launch

Display the graph for all to see.

A parabola on a coordinate plane.

Ask students to identify the intercepts and vertex. (vertex: \((2, \text{-}1)\), \(x\)-intercepts: \((1,0), (3,0)\), \(y\)-intercept: \((0,3)\))

Student Facing

  1. Graph the function \(y = x^2 -10x + 16\).
  2. Find the coordinates for the
    1. \(x\)-intercepts
    2. \(y\)-intercept
    3. vertex
  3. Draw a dashed line along the line of symmetry for the graph.
  4. What do you notice about the line of symmetry as it relates to the:
    1. vertex
    2. \(x\)-intercepts
  5. Use the line of symmetry and the \(y\)-intercept to find another point on the parabola.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is to identify ways to find intercepts, vertex, and line of symmetry for a quadratic function that is graphed. Select students to share their responses and methods. Ask students,

  • “For this graph, the vertex is on the line of symmetry. Will that always be the case? Explain your reasoning.” (Yes, the vertex is always on the line of symmetry. If the vertex was not on the line of symmetry, there would be 2 of them.)
  • “Is the line of symmetry always halfway between the \(x\)-intercepts? Explain your reasoning.” (As long as there are two \(x\)-intercepts, the line of symmetry will always be halfway between them. Since the intercepts are along the same horizontal line (the \(x\)-axis), the vertical line of symmetry should always be halfway between them.)

22.3: What Do You Know? (15 minutes)

Activity

In this activity, students gather what information they can about quadratic functions without graphing at all and with only looking at the graph. Students must look for and make use of structure (MP7) when they determine information about a quadratic graph without the \(y\)-axis labeled.

Monitor for students who find the \(y\)-intercept from the vertex form by:

  1. finding the standard form
  2. substituting 0 for \(x\) and solving for the \(y\)-coordinate of the intercept

Student Facing

  1. Write a function that is represented by a graph with \(x\)-intercepts at \((\text-3,0)\) and \((1,0)\).
    1. Without graphing the function, find the \(y\)-intercept. Explain or show your reasoning.
    2. Without using graphing technology, use the three points you know to sketch the graph of this equation.

      Blank coordinate grid, origin O. X and y axis from negative 8 to 8, by 2s.
    3. What is the \(x\)-coordinate of the vertex? Explain your reasoning.
    4. Using the \(x\)-coordinate you found for the vertex, find the coordinate pair for the vertex.
  2. Graph on x and y axis. Graph goes through points negative 2 comma 0 and 4 comma 0.
    1. What do you know about the coordinates of the \(y\)-intercept?
    2. What do you know about the coordinates of the vertex?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of the discussion is to determine methods for finding important points from the equation as well as from the graph. Select students to share their solutions, including previously identified students. Ask students,

  • “Is the \(x\)-intercept easier to find with a function in factored form, like \(y = (x-1)(x+2)\), or in standard form, like \(y = x^2 + x - 2\)? Which is easier to find the \(y\)-intercept? Which is easier to find the vertex?” (The \(x\)-intercepts are easiest to find in the factored form. The \(y\)-intercept is easiest to find from the standard form. The vertex is easiest to find from the factored form since it is halfway between the \(x\)-intercepts which is easiest to find from the factored form.)
  • “What could be added to the graph to improve the information you know about the coordinates for the \(y\)-intercept and vertex?” (A grid or at least marks along the axis would help improve estimates for the coordinates.)