# Changing the Vertex

Illustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Standards: HSF-IF.CHSF-BF.B.3HSF-IF.C.7.a

18 questions

# Changing the Vertex

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##### Graphs of Two Functions

Here are graphs representing the functions ﻿$f$﻿ and ﻿$g$﻿, given by ﻿$f(x) = x(x + 6)$﻿ and ﻿$g(x) = x(x + 6) + 4$﻿.

1) Which graph represents function ﻿$f$﻿?

2) Which graph represents function ﻿$g$﻿?

3) Explain how you know.

4) Where does the graph of ﻿$f$﻿ meet the ﻿$x$﻿-axis?

5) Explain how you know.

##### Shifting the Graph

6) How would you change the equation ﻿$y = x^2$﻿ so that the vertex of the graph of the new equation is located at (0, 11) and the graph opens upward?

Show Work

7) How would you change the equation ﻿$y = x^2$﻿ so that the vertex of the graph of the new equation is located at (7, 11) and the graph opens upward?

Show Work

8) How would you change the equation ﻿$y = x^2$﻿ so that the vertex of the graph of the new equation is located at (7, -3) and the graph opens downward?

Show Work

Kiran graphed the equation ﻿$y = x^2 + 1$﻿ and noticed that the vertex is at (0, 1). He changed the equation to ﻿$y = (x - 3)^2 + 1$﻿ and saw that the graph shifted 3 units to the right and the vertex is now at (3, 1).

Next, he graphed the equation ﻿$y = x^2 + 2x + 1$﻿, observed that the vertex is at (-1, 0). Kiran thought, “If I change the squared term ﻿$x^2$﻿ to ﻿$(x - 5)^2$﻿, the graph will move 5 units to the right and the vertex will be at (4, 0).”

9) Do you agree with Kiran?

True or false? Write below.

10) Explain or show your reasoning.

##### A Peanut Jumping over a Wall

Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to show the path of the main character, an animated peanut, jumping over a wall.

Mai uses the equation ﻿$y = -0.1(x - h)^2 + k$﻿ to represent the path of the jump. ﻿$y$﻿ represents the height of the peanut as a function of the horizontal distance it travels . On the screen, the base of the wall is located at (22, 0), with the top of the wall at (22, 4.5).

The dashed curve in the picture shows the graph of one equation Mai tried, where the peanut fails to make it over the wall.

11) What is the value of ﻿$k$﻿ in this equation?

12) What is the value of ﻿$h$﻿ in this equation?

##### Smiley Face

Do you see 2 “eyes” and a smiling “mouth” on the graph? The 3 arcs on the graph all represent quadratic functions that were initially defined by ﻿$y = x^2$﻿, but whose equations were later modified.

13) Write an equation to represent the left eye in the smiley face.

Show Work

14) What is the domain used for this function?

15) Write an equation to represent the right eye in the smiley face.

Show Work

16) What is the domain used for this function?

17) Write an equation to represent the mouth in the smiley face.

Show Work

18) What is the domain used for this function?