Question

1. describe the transformations from f(x) = -x - 6 and g(x) = 3x - 1. select all that apply. reflected over the y - axis. three times as steep translated up 5 units one - third as steep translated down 5 units

Explanation:

Step1: Analyze slope transformation

The slope of \( f(x)= -x - 6 \) is \( m_f=- 1 \), and the slope of \( g(x)=3x - 1 \) is \( m_g = 3 \). The absolute value of the slope of \( g(x) \) is \( |3|=3 \) and of \( f(x) \) is \( | - 1| = 1 \). Since \( 3\div1=3 \), the line \( g(x) \) is three times as steep as \( f(x) \) (considering the steepness is related to the absolute value of the slope).

Step2: Analyze vertical translation

The y - intercept of \( f(x) \) is \( b_f=-6 \) and of \( g(x) \) is \( b_g=-1 \). To find the vertical translation, we calculate \( b_g - b_f=-1-(-6)=-1 + 6 = 5 \). A positive value means the graph of \( g(x) \) is translated up 5 units from the graph of \( f(x) \).

Step3: Check reflection over y - axis

A reflection over the y - axis of a function \( y = f(x) \) gives \( y=f(-x) \). For \( f(x)=-x - 6 \), \( f(-x)=-(-x)-6=x - 6 \), which is not equal to \( g(x)=3x - 1 \), so there is no reflection over the y - axis. Also, the slope of \( g(x) \) is not \( \frac{1}{3}\) of the slope of \( f(x) \) (since \( \frac{1}{3}\times| - 1|=\frac{1}{3}
eq3 \)) and the vertical translation is up 5 units, not down.

Answer:

• Three times as steep
• Translated up 5 units