Question

A function is represented by the graph.

Complete the statement by selecting from the drop-down menu.

The y-intercept of the function y = 2x + 1 ​​is
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the y-intercept of the function represented in the graph.
A graph with a line running through point (-3, -1) and point (6, 5)

Answer 1

To find the y-intercept of the function \( y = 2x + 1 \), we evaluate the function when \( x = 0 \):

\[
y = 2(0) + 1 = 1.
\]

Thus, the y-intercept of the function \( y = 2x + 1 \) is \( (0, 1) \).

Now, let's analyze the line running through the points (-3, -1) and (6, 5) in the graph.

First, we can find the slope of the line using the formula:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-1)}{6 - (-3)} = \frac{5 + 1}{6 + 3} = \frac{6}{9} = \frac{2}{3}.
\]

Next, we can use one of the points (let's take -3, -1) to find the equation of the line using the point-slope form \( y - y_1 = m(x - x_1) \):

\[
y - (-1) = \frac{2}{3}(x - (-3)) \implies y + 1 = \frac{2}{3}(x + 3).
\]

To convert this to slope-intercept form, distribute and simplify:

\[
y + 1 = \frac{2}{3}x + 2 \implies y = \frac{2}{3}x + 2 - 1 \implies y = \frac{2}{3}x + 1.
\]

Now we can find the y-intercept of this line:

For \( x = 0 \):

\[
y = \frac{2}{3}(0) + 1 = 1,
\]

which gives the y-intercept at \( (0, 1) \).

Both functions have the same y-intercept of \( 1 \).

Thus, you can complete the statement as follows:

The y-intercept of the function \( y = 2x + 1 \) is **equal to** the y-intercept of the function represented in the graph.