To find a linear function $h(x) = mx + b$, we need to determine the slope $m$ and the y-intercept $b$.

We are given two points: $h(3) = -5$ and $h(-4) = 23$. These points can be written as $(3, -5)$ and $(-4, 23)$.

Step 1: Find the slope $m$

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the given points $(3, -5)$ and $(-4, 23)$:

$m = \frac{23 - (-5)}{-4 - 3} = \frac{23 + 5}{-4 - 3} = \frac{28}{-7} = -4$

Step 2: Find the y-intercept $b$

Using the slope $m = -4$ and one of the points, say $(3, -5)$, we can plug these into the linear equation $h(x) = mx + b$ to solve for $b$.

$-5 = -4(3) + b$ $-5 = -12 + b$ $b = -5 + 12 = 7$

Step 3: Write the equation of the linear function

Now that we have the slope $m = -4$ and the y-intercept $b = 7$, the equation of the linear function is:

$h(x) = -4x + 7$

Part 2: Find $h(5)$

Now, substitute $x = 5$ into the linear function $h(x) = -4x + 7$:

$h(5) = -4(5) + 7 = -20 + 7 = -13$

Final Answer:

• The linear function is $h(x) = -4x + 7$.
• $h(5) = -13$.

Would you like more details on any part of the solution or have additional questions?

Here are some related questions to explore:

1. How do you find the slope from any two points on a line?
2. How do changes in the slope affect the graph of a linear function?
3. How can we graph a linear function step-by-step?
4. What is the significance of the y-intercept in a linear equation?
5. How can we quickly verify if a point lies on the graph of a given linear function?

Tip: Always simplify slope and intercept calculations to avoid errors in finding the final function.