Question

which function is represented by the graph below?
options:
y = -4x + \frac{1}{2}
y = -4x + 2
y = \frac{1}{2}x - 4
y = 2x - 4

Explanation:

Step1: Analyze the slope

The graph is a straight line, so it's a linear function in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.
We can see from the graph that when \( x = 2 \), \( y = 0 \) and when \( x = 4 \), \( y = 4 \).
The formula for slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let \( (x_1,y_1)=(2,0) \) and \( (x_2,y_2)=(4,4) \). Then \( m=\frac{4 - 0}{4 - 2}=\frac{4}{2}=2 \). So the slope \( m = 2 \).

Step2: Analyze the y - intercept

The y - intercept \( b \) is the value of \( y \) when \( x = 0 \). From the graph, when \( x = 0 \), \( y=-4 \).
Now let's check the equations:

• For \( y=-4x+\frac{1}{2} \), slope \( m=-4

eq2 \), so this is not the function.

• For \( y = - 4x+2 \), slope \( m=-4

eq2 \), so this is not the function.

• For \( y=\frac{1}{2}x - 4 \), slope \( m=\frac{1}{2}

eq2 \), so this is not the function.

• For \( y = 2x-4 \), slope \( m = 2 \) and when \( x = 0 \), \( y=-4 \), which matches the slope and y - intercept we found from the graph.

Answer:

\( y = 2x-4 \)