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To find the equation of the linear function in the form \( y = mx + b \), we can use the given information:

We are told that:
- \( f(5) = 0 \) (i.e., when \( x = 5 \), \( y = 0 \))
- \( f(0) = 4 \) (i.e., when \( x = 0 \), \( y = 4 \))

### Step 1: Start with the equation of a line \( y = mx + b \).

Since \( f(0) = 4 \), this tells us that the \( y \)-intercept is 4. Thus, \( b = 4 \). The equation now becomes:
\[
y = mx + 4
\]

### Step 2: Use the second point \( f(5) = 0 \).

Substitute \( x = 5 \) and \( y = 0 \) into the equation to find the slope \( m \):
\[
0 = m(5) + 4
\]
\[
0 = 5m + 4
\]
\[
5m = -4
\]
\[
m = -\frac{4}{5}
\]

### Step 3: Write the final equation.

Now that we know the slope \( m = -\frac{4}{5} \) and the intercept \( b = 4 \), the equation of the line is:
\[
f(x) = -\frac{4}{5}x + 4
\]

Thus, the equation of the linear function is:
\[
f(x) = -\frac{4}{5}x + 4
\]

Would you like more details on any part of the solution? 

Here are 5 related questions to deepen your understanding:
1. How would the equation change if the point \( f(5) \) was not zero?
2. How do you find the slope if you're given two points in general?
3. What is the significance of the \( y \)-intercept in a linear equation?
4. How can you verify the solution by substituting back the points into the equation?
5. Can you solve for the equation if only the slope is given?

**Tip:** The slope of a line between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Write the equation of the linear function in the form y = mx + b given the following information: f(5) = 0, f(0) = 4.

Learn how to find the equation of a linear function in the form y = mx + b using the points f(5)=0 and f(0)=4. This problem covers key algebraic concepts, including the slope-intercept form and slope calculation. Suitable for students in Grades 8-10.

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