if f(x) is a linear function, f(-5)=0 and f(2)=1, find an equation for f(x)

Given that \( f(x) \) is a linear function, its equation can be written in the form:

\[
f(x) = mx + b
\]

where \( m \) is the slope and \( b \) is the y-intercept.

### Step 1: Find the slope (\( m \))
The slope of a linear function is given by the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

We are given two points: \( (-5, 0) \) and \( (2, 1) \). Substituting these values into the slope formula:

\[
m = \frac{1 - 0}{2 - (-5)} = \frac{1}{7}
\]

So, the slope \( m = \frac{1}{7} \).

### Step 2: Find the y-intercept (\( b \))
Now that we know the slope, we can use one of the given points to find the y-intercept \( b \). Let's use the point \( (2, 1) \).

Substitute \( m = \frac{1}{7} \), \( x = 2 \), and \( f(x) = 1 \) into the equation \( f(x) = mx + b \):

\[
1 = \frac{1}{7}(2) + b
\]

\[
1 = \frac{2}{7} + b
\]

Now, solve for \( b \):

\[
b = 1 - \frac{2}{7} = \frac{7}{7} - \frac{2}{7} = \frac{5}{7}
\]

### Step 3: Write the equation of the function
Now that we have both \( m \) and \( b \), the equation for \( f(x) \) is:

\[
f(x) = \frac{1}{7}x + \frac{5}{7}
\]

Would you like further details or clarification?

Here are 5 related questions for further exploration:
1. How would the equation change if one of the points were different?
2. Can you express this function in terms of another point-slope form?
3. What does the graph of this function look like?
4. How do you find the x-intercept of this function?
5. How would you calculate the slope if the function were non-linear?

**Tip:** For any linear function, the slope represents the rate of change between two points.

If f(x) is a linear function, f(-5)=0 and f(2)=1, find an equation for f(x).

Learn how to find the equation of a linear function f(x) given two points: f(-5)=0 and f(2)=1. This problem covers important algebraic concepts such as calculating the slope and determining the y-intercept. Suitable for students in Grades 7-9.

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