# Lesson 4

Reflecting Functions

• Let’s reflect some graphs.

### Problem 1

The dashed function is the graph of $$f$$ and the solid function is the graph of $$g$$. Express $$g$$ in terms of $$f$$.

### Problem 2

The table gives some values of functions $$f$$ and $$g$$.

Which of these equations could be true for all values of $$x$$?

$$x$$ $$f(x)$$ $$g(x)$$
-2 4 $$\frac14$$
-1 2 $$\frac12$$
0 1 1
1 $$\frac12$$ 2
2 $$\frac14$$ 4
A:

$$f(x) = \text-g(x)$$

B:

$$f(x) = g(-x)$$

C:

$$f(x) = \text-g(-x)$$

D:

$$f(x) = g(x)$$

### Problem 3

Here is the graph of a function $$f$$.

1. On the same axis, sketch a graph of $$f$$ reflected over the $$y$$-axis and then translate it 3 units up.
2. Write an equation (in terms of $$f$$) for a function $$g$$ that has the graph that you drew.

### Problem 4

Describe a transformation of the line that contains the two labelled points.

(From Unit 5, Lesson 1.)

### Problem 5

The thermostat in an apartment is set to $$75^\circ \text{F}$$ while the owner is awake and to $$60^\circ \text{F}$$ while the owner is sleeping. The function $$W$$ gives the temperature $$W(x)$$, in degrees Fahrenheit, in the apartment $$x$$ hours after midnight. When it is hot outside, the owner changes the settings to be exactly 10 degrees warmer than $$W$$ to save energy. The function $$H$$ gives the temperature $$H(x)$$, in degrees Fahrenheit, $$x$$ hours after midnight when it is hot outside.

1. If $$W(6.5) = 75$$, then what is the corresponding point on $$H$$? Use function notation to describe the point on $$H$$.
2. If $$W(2) = 60$$, then what is the corresponding point on $$H$$? Use function notation to describe the point on $$H$$.
3. Write an expression for $$H$$ in terms of $$W$$.
(From Unit 5, Lesson 2.)

### Problem 6

A ball is hit in the air. Its height $$h$$, in feet, $$t$$ seconds after it is hit is modeled by the equation $$h = 4 + 50t - 32t^2$$. Which equation models the height of a ball following the same path but is hit 2 seconds after the first ball?

A:

$$h = 6 + 50t - 32t^2$$

B:

$$h = 2 + 50t - 32t^2$$

C:

$$h = 4 + 50(t+2) -32(t+2)^2$$

D:

$$h = 4 + 50(t-2) - 32(t-2)^2$$

(From Unit 5, Lesson 3.)