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Chapter 1: Problem 79

Write a rule for a linear function \(y=f(x)\), given that \(f(0)=4\) and\(f(3)=11\).

### Short Answer

Expert verified

The linear function is \[ y = \frac{7}{3}x + 4 \]

## Step by step solution

01

## - Identify the given points

The function passes through the points \(0, 4\) and \(3, 11\). These points will help to determine the equation of the linear function.

02

## - Find the slope (m)

Use the formula for the slope of a line, \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) = (0, 4) \) and \( (x_2, y_2) = (3, 11) \). \[ \text{slope} = \frac{11 - 4}{3 - 0} = \frac{7}{3} \]

03

## - Use the slope-intercept form

The slope-intercept form of a line is \( y = mx + b \). Here, \( m = \frac{7}{3} \). Using the point \( (0, 4) \), substitute into the equation \[4 = \frac{7}{3}(0) + b \therefore b = 4 \]

04

## - Write the rule for the linear function

Substitute the values of \( m \) and \( b \) into the slope-intercept form: \[ y = \frac{7}{3}x + 4 \]

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Slope-Intercept Form

The slope-intercept form is a way to write the equation of a straight line. It is written as:

\( y = mx + b \).

In this equation:

- \(m\) represents the slope of the line.
- \(b\) represents the y-intercept, which is where the line crosses the y-axis.

This form is useful because it gives you both the slope and the y-intercept directly. Knowing these lets you easily graph the line and understand its behavior.

For example, in the exercise, the final linear function is \( y = \frac{7}{3}x + 4 \). Here, \( \frac{7}{3} \) is the slope, and \( 4 \) is the y-intercept.

###### Slope Calculation

The slope of a line tells you how steep the line is. It's calculated as the 'rise' over the 'run' between two points:

\( \frac{y_2 - y_1}{x_2 - x_1} \).

In this exercise, you were given the points \((0, 4)\) and \((3, 11)\). Using the slope formula:

\( \text{slope} = \frac{11 - 4}{3 - 0} = \frac{7}{3} \).

This means for every 3 units you move to the right on the x-axis, you move up 7 units on the y-axis. Understanding how to calculate the slope is essential for finding the equation of a line and graphing it correctly.

###### Using Given Points

When you're given specific points that a line passes through, you can use them to find the slope and the y-intercept.

Here's how you used the given points in the exercise:

- First, you identified the points: \((0, 4)\) and \((3, 11)\).
- You calculated the slope \(m\) using these points.
- Then, you used the point \((0, 4)\) to find the y-intercept \(b\). Since \(x = 0\), the y-coordinate of this point is the y-intercept directly.

Finally, you applied these values to the slope-intercept form to get the linear function: \( y = \frac{7}{3}x + 4 \).

This process shows how powerful these formulas are in helping you quickly find the equation of any line.

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