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Chapter 2: Problem 87

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

Short Answer

Expert verified
The rule is \(y = \frac{7}{3}x + 4\).

Step by step solution

01

Understand the Problem

A linear function can be written in the form of \[y = mx + b\]where \(m\)\ is the slope and \(b\)\ is the y-intercept.
02

Find the Y-intercept

Given that \(f(0) = 4\), it means that when \(x = 0\), \(y = 4\). So, \(b = 4\)
03

Use Given Points to Find the Slope

The slope \(m\) can be found using the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\]With the points \((x_1, y_1) = (0, 4)\) and \((x_2, y_2) = (3, 11)\), substitute these values into the formula to get:\[m = \frac{11 - 4}{3 - 0} = \frac{7}{3}\]
04

Write the Rule for the Linear Function

Now, substituting the slope \(m = \frac{7}{3}\) and y-intercept \(b = 4\) into the linear function form \[y = mx + b\], gives:\[y = \frac{7}{3}x + 4\]

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Key Concepts

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

Slope Calculation
To determine the slope of a linear function, you need two points on the line. The slope, often represented as \(m\), measures the steepness or incline of the line. It is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula is derived from the idea that the slope is the ratio of the 'rise' to the 'run' between two points.
  • 'Rise' refers to the change in the y-values (vertically).
  • 'Run' refers to the change in the x-values (horizontally).

Let's apply this formula to our problem. The two points given are \((0, 4)\) and \((3, 11)\). Plugging these points into the formula we get:
\[m = \frac{11 - 4}{3 - 0} = \frac{7}{3} \].
So, the slope of our linear function is \(\frac{7}{3}\).
Y-Intercept
The y-intercept of a linear function is the point where the line crosses the y-axis. In the equation form \(y = mx + b\), the y-intercept is \(b\). It shows the value of y when x is zero. To find it, we look at the function given at \(f(0)\).
For our problem, we are given that \(f(0) = 4\). This directly means that when \(x = 0\), \(y = 4\). Therefore, our y-intercept is \(b = 4\).
The y-intercept tells us where the line starts on the graph if we are considering it from the y-axis. It's a crucial component as it provides the starting point of the line on the graph.
Linear Equation
A linear equation describes a straight line on a graph and has the general form \(y = mx + b\), where
  • \(m\) is the slope of the line
  • \(b\) is the y-intercept

Using the slope and y-intercept we calculated earlier, we can construct the equation. We found that the slope \(m\) is \(\frac{7}{3}\) and the y-intercept \(b\) is 4.
So, the linear equation becomes:
\[ y = \frac{7}{3}x + 4 \].
This equation means that for any value of x, you can find the corresponding value of y by applying the formula. The linear equation is essential because it allows you to predict the values of y for given x-values, forming a consistent straight line when plotted on a graph.:

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Most popular questions from this chapter

Explain what it means for a function to be increasing on an interval.

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