Question

consider this table of values.
input | 0 | 2 | 4 | 6
output | 1 | 3 | 5 | 7
which of these graphs corresponds to the function shown in the table of values?
four graphs labeled w, x, y, z are shown
options:
a. x
b. y
c. z
d. w

Explanation:

Step1: Analyze the table

The table gives input (x) and output (y) values: when \( x = 0 \), \( y = 1 \); \( x = 2 \), \( y = 3 \); \( x = 4 \), \( y = 5 \); \( x = 6 \), \( y = 7 \). We can find the equation of the line. The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3 - 1}{2 - 0}=\frac{2}{2}=1 \). Using the slope - intercept form \( y=mx + b \), when \( x = 0 \), \( y = 1 \), so \( b = 1 \). The equation is \( y=x + 1 \).

Step2: Check the y - intercept

The y - intercept of the line \( y=x + 1 \) is 1 (when \( x = 0 \), \( y = 1 \)). Now we check the graphs:

• Graph W: The line passes through the origin (\( y = 0 \) when \( x = 0 \)), so it is not correct.
• Graph X: The y - intercept is 1? No, when \( x = 0 \), the y - value is 1? Wait, looking at the grid, Graph X seems to have a y - intercept of 1? Wait, no, let's check the slope. For Graph X, when \( x = 0 \), \( y = 1 \)? Wait, no, the second graph (X) has a line that starts at (0,1)? Wait, no, let's recalculate the slope for each graph.
• Graph Z: When \( x = 0 \), \( y = 1 \) (matches the y - intercept of our line \( y=x + 1 \)). When \( x = 2 \), \( y=2 + 1=3 \) (matches the table). When \( x = 4 \), \( y=4 + 1 = 5 \) (matches the table). When \( x = 6 \), \( y=6 + 1=7 \) (matches the table).
• Graph Y: The line passes through the origin (\( y = 0 \) when \( x = 0 \)), so it is not correct.
• Graph W: Passes through the origin, incorrect.
• Graph X: Let's check the slope. If we take two points on Graph X, say (0,1) and (2,2), the slope is \( \frac{2 - 1}{2 - 0}=\frac{1}{2}

eq1 \), so it is incorrect.

• Graph Z: The slope is 1 (since for every increase of 1 in x, y increases by 1) and the y - intercept is 1, which matches our equation \( y=x + 1 \) and the table values.

Answer:

C. Z