Question

1. write a linear equation for each function. which function has the greater y - intercept? explain your reasoning.

the table has x and y columns. when x is -2, y is -20; when x is 0, y is -4; when x is 2, y is 12; when x is 4, y is 28. there is also a graph of a line on a coordinate plane with grid lines and axes labeled with 0, 5, -5.

Explanation:

For the table - based function:

Step1: Find the slope ($m$)

We use two points from the table, say $(0, - 4)$ and $(2,12)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
So, $m=\frac{12-(-4)}{2 - 0}=\frac{16}{2}=8$.

Step2: Find the y - intercept ($b$)

The y - intercept is the value of $y$ when $x = 0$. From the table, when $x = 0$, $y=-4$, so $b=-4$.

Step3: Write the linear equation

Using the slope - intercept form $y=mx + b$, the equation is $y = 8x-4$.

For the graph - based function:

Step1: Identify two points on the graph

From the graph, we can see that the line passes through $(0, - 5)$ (the y - intercept) and $(5,0)$.

Step2: Find the slope ($m$)

Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ with $(0,-5)$ and $(5,0)$:
$m=\frac{0 - (-5)}{5-0}=\frac{5}{5}=1$.

Step3: Write the linear equation

Using the slope - intercept form $y = mx + b$, where $b=-5$ (from the point $(0,-5)$) and $m = 1$, the equation is $y=x - 5$.

Comparing the y - intercepts:

The y - intercept of the table - based function is $-4$, and the y - intercept of the graph - based function is $-5$. Since $-4>-5$, the function represented by the table has the greater y - intercept.

Answer:

The linear equation for the table - based function is $y = 8x-4$, for the graph - based function is $y=x - 5$. The function represented by the table (with equation $y = 8x-4$) has the greater y - intercept because its y - intercept $(-4)$ is greater than the y - intercept of the graph - based function $(-5)$.