Questions: How the leading coefficient affects the graph of an absolute value function Fill in the information about the absolute value functions below. y=1/2x, y=2/3x, y=2x, y=-4x (a) For each function, choose whether its graph opens upward or downward. y=1/2x: (Choose one) ^ y=2/3x: (Choose one) ^ y=2x: (Choose one) ^ y=-4x: (Choose one) i (b) Choose the equation with the widest graph. y=1/2x y=2/3x y=2x y=-4x (c) Choose the equation with the narrowest graph. y=1/2x y=2/3x y=2x y=-4x

How the leading coefficient affects the graph of an absolute value function

Fill in the information about the absolute value functions below.

y=1/2x, y=2/3x, y=2x, y=-4x

(a) For each function, choose whether its graph opens upward or downward.
y=1/2x: (Choose one) ^ y=2/3x: (Choose one) ^ y=2x: (Choose one) ^ y=-4x: (Choose one) i

(b) Choose the equation with the widest graph.
y=1/2x y=2/3x y=2x y=-4x

(c) Choose the equation with the narrowest graph.
y=1/2x y=2/3x y=2x y=-4x

Transcript text: How the leading coefficient affects the graph of an absolute value function Fill in the information about the absolute value functions below. \[ y=\frac{1}{2}|x|, \quad y=\frac{2}{3}|x|, \quad y=2|x|, \quad y=-4|x| \] (a) For each function, choose whether its graph opens upward or downward. \[ \left.\left.\left.y=\frac{1}{2}|x|:(\text { Choose one })^{\vee} \quad y=\frac{2}{3}|x|: \text { (Choose one }\right)^{\vee} \quad y=2|x|: \text { (Choose one }\right)^{\vee} \quad y=-4|x|: \text { (Choose one }\right)^{\imath} \] (b) Choose the equation with the widest graph. $y=\frac{1}{2}|x|$ $y=\frac{2}{3}|x|$ $y=2|x|$ $y=-4|x|$ (c) Choose the equation with the narrowest graph. $y=\frac{1}{2}|x|$ $y=\frac{2}{3}|x|$ $y=2|x|$ $y=-4|x|$

Solution

Solution Steps

Step 1: Determine the direction of the graph (upward or downward)

For $ y = \frac{1}{2}|x| $, the leading coefficient $ \frac{1}{2} $ is positive, so the graph opens upward.
For $ y = \frac{2}{3}|x| $, the leading coefficient $ \frac{2}{3} $ is positive, so the graph opens upward.
For $ y = 2|x| $, the leading coefficient $ 2 $ is positive, so the graph opens upward.
For $ y = -4|x| $, the leading coefficient $ -4 $ is negative, so the graph opens downward.

Step 2: Identify the equation with the widest graph

The width of the graph of an absolute value function is inversely proportional to the absolute value of the leading coefficient.
The smallest leading coefficient in absolute value is $ \frac{1}{2} $, so $ y = \frac{1}{2}|x| $ has the widest graph.

Step 3: Identify the equation with the narrowest graph