# Eureka Math Algebra 1 Module 3 Lesson 19 Answer Key

## Engage NY Eureka Math Algebra 1 Module 3 Lesson 19 Answer Key

### Eureka Math Algebra 1 Module 3 Lesson 19 Exploratory Challenge Answer Key

Exploratory Challenge 1
Let f(x) = x2 and g(x) = f(2x), where x can be any real number.
a. Write the formula for g in terms of x2 (i.e., without using f(x) notation).
g(x) = (2x)2

b. Complete the table of values for these functions.

c. Graph both equations: y = f(x) and y = f(2x).

See the discussion below for an explanation of the steps and arrows.

d. How does the graph of y = g(x) relate to the graph of y = f(x)?
The corresponding x-value of y = g(x) is half of the corresponding x-value of y = f(x) when g(x) = f(x); the points of the graph of g are $$\frac{1}{2}$$ the distance to the y-axis as the corresponding points of the graph of f, which makes the graph of g appear to “shrink horizontally.”

e. How are the values of f related to the values of g?
For equal outputs of f and g, the input of g only has to be half as big as the input of f.

Exploratory Challenge 2
Let f(x) = x2 and h(x) = f($$\frac{1}{2}$$ x), where x can be any real number.
a. Rewrite the formula for h in terms of x2 (i.e., without using f(x) notation).
h(x) = ($$\frac{1}{2}$$ x)2

b. Complete the table of values for these functions.

c. Graph both equations: y = f(x) and y = f($$\frac{1}{2}$$ x).

d. How does the graph of y = f(x) relate to the graph of y = h(x)?
Since the corresponding x-value of y = h(x) is twice the corresponding x-value of y = f(x) when h(x) = f(x), the points of the graph of h are 2 times the distance to the y-axis as the corresponding points of the graph of f, which makes the graph of h appear to “stretch horizontally.”

e. How are the values of f related to the values of h?
To get equal outputs of each function, the input of h has to be twice the input of f.

Exploratory Challenge 3
a. Look at the graph of y = f(x) for the function f(x) = x2 in Exploratory Challenge 1 again. Would we see a difference in the graph of y = g(x) if -2 were used as the scale factor instead of 2? If so, describe the difference. If not, explain why not.
There would be no difference. The function involves squaring the value within the parentheses; so, both the graph of y = f(2x) and the graph of y = f(-2x) are the same set as the graph of y = g(x), but both correspond to different transformations: The first is a horizontal scaling with scale factor $$\frac{1}{2}$$, and the second is a horizontal scaling with scale factor $$\frac{1}{2}$$ and a reflection across the y-axis.

b. A reflection across the y-axis takes the graph of y = f(x) for the function f(x) = x2 back to itself. Such a transformation is called a reflection symmetry. What is the equation for the graph of the reflection symmetry of the graph of y = f(x)?
y = f(-x).

c. Deriving the answer to the following question is fairly sophisticated; do this only if you have time. In Lessons 17 and 18, we used the function f(x) = |x| to examine the graphical effects of transformations of a function. In this lesson, we use the function f(x) = x2 to examine the graphical effects of transformations of a function. Based on the observations you made while graphing, why would using f(x) = x2 be a better option than using the function f(x) = |x|?
Not all of the effects of multiplying the input of a function are as visible with an absolute function as they are with a quadratic function. For example, the graph of y = 2|x| is the same as y = |2x|. Therefore, it is easier to see the effect of multiplying a value to the input of a function by using a quadratic function than it is by using the absolute value function.

### Eureka Math Algebra 1 Module 3 Lesson 19 Exercise Answer Key

Exercise
Complete the table of values for the given functions.
a.

b. Label each of the graphs with the appropriate functions from the table.

c. Describe the transformation that takes the graph of y = f(x) to the graph of y = g(x).
The graph of y = g(x) is a horizontal scale with scale factor $$\frac{1}{2}$$ of the graph of y = f(x).

d. Consider y = f(x) and y = h(x). What does negating the input do to the graph of f?
The graph of h is a reflection over the y-axis of the graph of f.

e. Write the formula of an exponential function whose graph would be a horizontal stretch relative to the graph of g.
Answers vary. Example: y = 2(0.5x).

### Eureka Math Algebra 1 Module 3 Lesson 19 Problem Set Answer Key

Let f(x) = x2, g(x) = 2x2, and h(x) = (2x)2, where x can be any real number. The graphs above are of the functions y = f(x), y = g(x), and y = h(x).
Question 1.
Label each graph with the appropriate equation.

Question 2.
Describe the transformation that takes the graph of y = f(x) to the graph of y = g(x). Use coordinates to illustrate an example of the correspondence.
The graph of y = g(x) is a vertical stretch of the graph of y = f(x) by scale factor 2; for a given x-value, the value of g(x) is twice as much as the value of f(x).
OR
The graph of y = g(x) is a horizontal shrink of the graph of y = f(x) by scale factor $$\frac{1}{\sqrt{2}}$$. It takes $$\frac{1}{\sqrt{2}}$$ times the input for y = g(x) as compared to y = f(x) to yield the same output.

Question 3.
Describe the transformation that takes the graph of y = f(x) to the graph of y = h(x). Use coordinates to illustrate an example of the correspondence.
The graph of y = h(x) is a horizontal shrink of the graph of y = f(x) by a scale factor of $$\frac{1}{2}$$. It takes $$\frac{1}{2}$$ the input for y = h(x), as compared to y = f(x) to yield the same output.
OR
The graph of y = h(x) is a vertical stretch of the graph of y = f(x) by scale factor 4; for a given x-value, the value of h(x) is four times as much as the value of f(x).

### Eureka Math Algebra 1 Module 3 Lesson 19 Exit Ticket Answer Key

Let f(x) = x2, g(x) = (3x)2, and h(x) = ($$\frac{1}{3}$$ x)2, where x can be any real number. The graphs above are of y = f(x), y = g(x), and y = h(x).
Question 1.
Label each graph with the appropriate equation.
The graph of y = g(x) is a horizontal shrink of the graph of y = f(x) with scale factor $$\frac{1}{3}$$. The corresponding x-value of y = g(x) is one-third of the corresponding x-value of y = f(x) when g(x) = f(x). This can be illustrated with the coordinate (1, 9) on g(x) and the coordinate (3, 9) on f(x).