To determine which function represents the same relationship as the points given in the table, we first need to find the equation of the linear function that fits these points. Let's go through the process step-by-step:
1. Identify the Given Points:
The points provided in the table are:
- (-7.5, 12)
- (-3.5, 0)
- (-1, -7.5)
- (2, -16.5)
- (3.5, -21)
2. Calculate the Slope (m):
The slope of a linear function can be found using any two points \((x_1, y_1)\) and \((x_2, y_2)\). We’ll use the points (-7.5, 12) and (-3.5, 0).
The formula for the slope \(m\) is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the values:
[tex]\[
m = \frac{0 - 12}{-3.5 - (-7.5)} = \frac{-12}{4} = -3
\][/tex]
So, the slope \(m\) is \(-3\).
3. Find the Y-intercept (b):
Using the slope \(m\) and one of the points, we can find the y-intercept \(b\). We use the point \((-7.5, 12)\).
The formula for the y-intercept is derived from the equation of the line \(y = mx + b\):
[tex]\[
b = y - mx
\][/tex]
Plugging in the values:
[tex]\[
b = 12 - (-3)(-7.5) = 12 - 22.5 = -10.5
\][/tex]
So, the y-intercept \(b\) is \(-10.5\).
4. Write the Equation of the Line:
With \(m = -3\) and \(b = -10.5\), the equation of the linear function is:
[tex]\[
h(x) = -3x - 10.5
\][/tex]
5. Verification:
We verify this equation by checking if it correctly calculates the \(y\)-values for all the given \(x\)-points:
- For \(x = -7.5\):
[tex]\[
h(-7.5) = -3(-7.5) - 10.5 = 22.5 - 10.5 = 12
\][/tex]
- For \(x = -3.5\):
[tex]\[
h(-3.5) = -3(-3.5) - 10.5 = 10.5 - 10.5 = 0
\][/tex]
- For \(x = -1\):
[tex]\[
h(-1) = -3(-1) - 10.5 = 3 - 10.5 = -7.5
\][/tex]
- For \(x = 2\):
[tex]\[
h(2) = -3(2) - 10.5 = -6 - 10.5 = -16.5
\][/tex]
- For \(x = 3.5\):
[tex]\[
h(3.5) = -3(3.5) - 10.5 = -10.5 - 10.5 = -21
\][/tex]
The computed \(y\)-values match the given \(y\)-values for all points.
6. Conclusion:
Comparing the equation we derived, \(h(x) = -3x - 10.5\), with the given options, we find that Option A correctly represents the same relationship.
Final Answer:
[tex]\[
\boxed{A. \ h(x) = -3x - 10.5}
\][/tex]
