The two points on the graph are given by the linear function f.
a. Use the two points to find the equation that represents the linear function f.
b. What is​ f(6)?
-1
1
2
3
4
5
-1
1
2
3
x
y



A coordinate system has a horizontal x-axis labeled from negative 1 to 5 in increments of 0.5 and a vertical y-axis labeled from negative 1 to 3 in increments of 0.5. Points are plotted at (1, 2) and (4, 1).
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Part 1
a. The equation ​f(x)equals
  
enter your response here represents the linear function f.

To find the equation of the linear function \( f(x) \) passing through the points \( (1, 2) \) and \( (4, 1) \), we'll use the slope-intercept form of a line:

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

### Step 1: Calculate the Slope \( m \)
The slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Plugging in the points \( (1, 2) \) and \( (4, 1) \):
\[
m = \frac{1 - 2}{4 - 1} = \frac{-1}{3} = -\frac{1}{3}
\]

### Step 2: Use the Slope to Find the y-Intercept \( b \)
Now that we know \( m = -\frac{1}{3} \), we substitute one of the points into the equation \( f(x) = mx + b \) to solve for \( b \).

Using the point \( (1, 2) \):
\[
2 = -\frac{1}{3}(1) + b
\]
\[
2 = -\frac{1}{3} + b
\]
\[
b = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}
\]

### Step 3: Write the Equation
The equation of the line is:
\[
f(x) = -\frac{1}{3}x + \frac{7}{3}
\]

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### Part (b): Calculate \( f(6) \)
To find \( f(6) \), substitute \( x = 6 \) into the equation:
\[
f(6) = -\frac{1}{3}(6) + \frac{7}{3}
\]
\[
f(6) = -2 + \frac{7}{3}
\]
Converting \(-2\) to a fraction:
\[
f(6) = -\frac{6}{3} + \frac{7}{3} = \frac{1}{3}
\]

### Final Answers
a. The equation for \( f(x) \) is:
\[
f(x) = -\frac{1}{3}x + \frac{7}{3}
\]

b. \( f(6) = \frac{1}{3} \)

Would you like further details or have questions on any steps?

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Here are 5 related questions for further practice:
1. What is the slope of a line that passes through points \((2, 4)\) and \((5, 7)\)?
2. If \( f(x) = -\frac{1}{3}x + \frac{7}{3} \), find \( f(-3) \).
3. How do you determine if a function is linear by looking at its equation?
4. What is the equation of a line with a slope of \( \frac{2}{5} \) that passes through \((0, -3)\)?
5. Find the x-intercept and y-intercept of the equation \( f(x) = -\frac{1}{3}x + \frac{7}{3} \).

**Tip:** When finding the slope of a line between two points, remember that the formula is \((y_2 - y_1) / (x_2 - x_1)\) to ensure consistency in direction.

The two points on the graph are given by the linear function f.
a. Use the two points to find the equation that represents the linear function f.
b. What is f(6)?

A coordinate system has a horizontal x-axis labeled from -1 to 5 in increments of 0.5 and a vertical y-axis labeled from -1 to 3 in increments of 0.5. Points are plotted at (1, 2) and (4, 1).

Discover how to find the linear function equation that passes through points (1, 2) and (4, 1), and learn how to calculate f(6) using the slope-intercept form. Includes key algebra concepts and step-by-step solutions for students in Grades 8-10.

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