To determine the equation of a linear function from a table of values, we need to find the slope and the y-intercept. Here is a step-by-step guide to solving this problem:
1. Identify two points from the table:
Let's select the points [tex]\((0, -5)\)[/tex] and [tex]\((1, -1)\)[/tex].
2. Calculate the slope ([tex]\(m\)[/tex]):
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting our points into this formula, we get:
[tex]\[
m = \frac{-1 - (-5)}{1 - 0} = \frac{-1 + 5}{1} = \frac{4}{1} = 4
\][/tex]
So, the slope [tex]\(m\)[/tex] is [tex]\(4.0\)[/tex].
3. Find the y-intercept ([tex]\(b\)[/tex]):
The y-intercept occurs where [tex]\(x = 0\)[/tex]. From the table, we can see that when [tex]\(x = 0\)[/tex], [tex]\(y = -5\)[/tex]. Hence, the y-intercept [tex]\(b\)[/tex] is [tex]\(-5.0\)[/tex].
4. Write the equation of the line:
The general form of the equation of a line is:
[tex]\[
y = mx + b
\][/tex]
Substituting the slope [tex]\(m = 4.0\)[/tex] and the y-intercept [tex]\(b = -5.0\)[/tex] into this formula, we get:
[tex]\[
y = 4.0x - 5.0
\][/tex]
So, the slope of the function's graph is [tex]\(4.0\)[/tex], the y-intercept is [tex]\(-5.0\)[/tex], and the equation of the function is:
[tex]\[
y = 4.0x - 5.0
\][/tex]
