Question

1. the graph of $y = -2|x + 1| - 9$ opens:

a) up
b) down
c) left
d) right

1. consider the graph of $y = |x|$. how would you shift the graph to sketch the function $f(x) = |x - 7| + 2$

a) left 7 and up 2
b) right 7 and down 2
c) left 7 and down 2
d) right 7 and up 2

1. sketch the graph of $y = \frac{1}{2}|x|$

grid graph for plotting

Explanation:

Question 3

Step1: Recall absolute value function form

The general form of an absolute value function is \( y = a|x - h| + k \), where \( a \) determines the direction and vertical stretch/compression.

Step2: Analyze the coefficient \( a \)

In the function \( y = -2|x + 1| - 9 \), the coefficient \( a = -2 \). If \( a > 0 \), the graph opens up; if \( a < 0 \), the graph opens down. Since \( -2 < 0 \), the graph opens down.

Step1: Recall horizontal shift rule

For a function \( y = |x - h| \), if \( h > 0 \), the graph shifts right by \( h \) units; if \( h < 0 \), it shifts left by \( |h| \) units. For vertical shift, \( y = |x| + k \) shifts up by \( k \) units if \( k > 0 \), down if \( k < 0 \).

Step2: Analyze \( f(x) = |x - 7| + 2 \)

In \( f(x) = |x - 7| + 2 \), \( h = 7 \) (so shift right 7 units) and \( k = 2 \) (shift up 2 units).

Step1: Recall parent function \( y = |x| \)

The parent function \( y = |x| \) has a V - shape with vertex at the origin, passing through \( (1,1) \) and \( (-1,1) \).

Step2: Analyze the transformation \( y=\frac{1}{2}|x| \)

The function \( y = \frac{1}{2}|x| \) is a vertical compression of the parent function \( y = |x| \) by a factor of \( \frac{1}{2} \). For the parent function, when \( x = 1 \), \( y = 1 \); for \( y=\frac{1}{2}|x| \), when \( x = 1 \), \( y=\frac{1}{2}(1)=\frac{1}{2} \), and when \( x=- 1 \), \( y = \frac{1}{2}(-1)\) (but absolute value makes it \( \frac{1}{2}\)). The vertex remains at \( (0,0) \), and the graph is a narrower (vertically compressed) V - shape compared to \( y = |x| \), opening upwards with a slope of \( \frac{1}{2} \) for \( x\geq0 \) and \( -\frac{1}{2} \) for \( x < 0 \). To sketch it, plot the vertex at \( (0,0) \), then plot points like \( (2,1) \) (since \( y=\frac{1}{2}\times|2| = 1 \)) and \( (-2,1) \), and draw the two lines of the V - shape.

Answer:

b) Down

Question 4