The Intercepts of a Quadratic Function

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Objective

SWBAT sketch a quadratic function using its intercepts.

Big Idea

Students will discover the relationship between the factors of a trinomial and its roots. Students will use the x and y axis to plot key points on a parabola.

Do-Now

10 minutes

Students will complete the Do Now in 5 minutes. We will then review the answers as a whole group. The task is simple, but it is intended to give my students a hint as to our use of intercepts in today's lesson.

Next, a student will read the objective, "SWBAT sketch a quadratic function using its intercepts".

Before we begin I will ask a student volunteer to summarize the information learned in our last class during the Quadratics Investigation.

Guided Notes + Investigation

40 minutes

In today's lesson students will graph quadratic functions on the coordinate plane without the use of a table. 

There are multiple processes involved in graphing quadratic functions. Rather than overwhelm students with too much too soon - today we will focus solely on the x and y axis. Expanding students knowledge of linear functions, we will apply our understanding of the coordinate plane to find the intercepts of the function. 

Using this handout, Page 1 will be introduced important terminology and  the standard form of a quadratic equation. I will model the identification of coefficients  A, B and C in a quadratic function. Additionally I will model how to transform these functions into standard form so that A B and C are easily recognizable. (Students practiced this skill during a previous lesson on Literal Equations).

Next, we will complete the investigation on Page 2 as a whole class. I will use use this website to display a graphing calculator on the board for the class to view. The graph is best displayed using the projector mode option found in settings.

Pairs will work individually to label and factor each trinomial. After a few minutes, students will share their responses aloud as a whole group.

Next, I will input the equation of each function on the screen. After graphing each example, I will ask students to meet careful observations about what is seen on the screen:

  • Where does this function cross the x-axis?
  • The place where a quadratic function crosses the x-axis are called caught its roots; do the roots have anything in common with the factors of the quadratic equation? What is it?
  • The opposite value of each factor is also the location of a root. Can anyone explain this? Why do you think this happens?
  • What do we know about all points that lie on the x-axis? 
  • (I will sketch a quick coordinate plane on the side of the board, then will plot three points to illustrate this question for students)
  • What conclusion can we make about all points that lie on the x-axis?
  • Each example begins with f(x). We learned in the beginning of the school year this notation also represents our y-value. Can you represent the location of the x-intercept using function notation?
  • We know that the graph of a quadratic function will cross the x-axis, and at the location the y value will always be zero. Can I  take each factor and set it equal to zero. What do you think the output corresponds to?
  • Where does this function cross the y-axis?
  • Have you seen this value somewhere else and this function?
  • What do we know about all points that lie on the y-axis? 
  • (If necessary, sketch a quick coordinate plane on the side of the board, then will plot three points to illustrate this question for students)
  • Why does it make sense that value of the constant term in a quadratic function will always be its y-intercept? Justify your response.

Next, students will work in pairs to find all intercepts in the example problems on Page 5. After 15 minutes we will review responses as a whole group. I will graph 6-7 example problems from this page using the Desmos calculator to solidify the relationship between the factors of a quadratic function and its intercepts.

Partner Practice - Matching

20 minutes

For additional practice with graphs of quadratic functions, my students will work with a partner to complete this Matching Activity. Each graph on Page 1 will match to a quadratic function on Page 2. This activity does not need to be cut up. On Page 2, students should identify and explain the x and y intercepts of each function inside of each box.

We will review this activity as a whole group towards the end of class.

Closing

10 minutes

To close today's lesson the class will complete the Closing Activity. This activity may seem redundant - but it is important that students enter our next class with the concepts learned during our investigation as background knowledge. If students are able to find the x and y intercepts of a quadratic function fluently, the addition of more components to the graph will not seem as daunting.  

We will review the responses as a whole group to the closing activity as a whole group. Students will then complete an exit card.