Lesson 22

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.

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.

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?”

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.

Display the graph for all to see.

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

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.)

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

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.)