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Chapter 1: Problem 25

In Exercises 17鈥�30, write an equation for each line described. Has \(y\) -intercept 4 and \(x\) -intercept \(-1\)

Short Answer

Expert verified
The equation is \( y = 4x + 4 \).

Step by step solution

01

Understanding the Problem

We are tasked with writing the equation of a line given its intercepts. The y-intercept is 4 and the x-intercept is -1.
02

Identifying the Form

To find the equation of the line, we can use the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
03

Applying Intercept Information

Given the intercepts, the points on the line are (0, 4) for the y-intercept and (-1, 0) for the x-intercept. These points can be used to calculate the slope.
04

Calculating the Slope

The slope \( m \) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1) = (0, 4)\) and \((x_2, y_2) = (-1, 0)\). Substituting the values, we get \( m = \frac{0 - 4}{-1 - 0} = \frac{-4}{-1} = 4 \).
05

Writing the Equation in Slope-Intercept Form

Now that we know the slope \( m = 4 \) and the y-intercept \( b = 4 \), we substitute these values into the slope-intercept form: \( y = 4x + 4 \).
06

Verification of Intercepts

To verify, substitute \( x = 0 \) to find \( y \), which gives \( y = 4 \), confirming the y-intercept. For the x-intercept, set \( y = 0 \) and solve \( 0 = 4x + 4 \), resulting in \( x = -1 \), confirming the x-intercept.

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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 express the equation of a straight line through its slope and y-intercept. This form is written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. It's a straightforward way to understand the general direction and position of a line on a coordinate plane.

The slope \( m \) tells us how steep the line is. A positive slope means the line ascends from left to right, while a negative slope means it descends. The value \( b \), the y-intercept, indicates where the line crosses the y-axis.

Using the slope-intercept form is extremely helpful in quickly graphing lines and understanding their behavior. By simply knowing the slope and y-intercept, you can sketch the graph of the line without needing any other points.
X-Intercept
The x-intercept of a line is the point where the line crosses the x-axis. This means the y-value at this point is always zero. To find the x-intercept, we set \( y = 0 \) in the line's equation and solve for \( x \).

Understanding the x-intercept is crucial since it provides insight into where the line interacts with the horizontal axis. In practical situations, x-intercepts can represent time, distance, or other variables reaching a baseline value.

In our exercise, the x-intercept is -1. We verified it by plugging \( y = 0 \) into the line equation \( y = 4x + 4 \), resulting in \( x = -1 \). This confirms where the line meets the x-axis.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis. Here, the x-value is always zero. To find the y-intercept in an equation, set \( x = 0 \) and solve for \( y \).

The y-intercept is an important aspect of the line's equation because it provides a starting point on the graph. It represents the value of \( y \) when \( x \) is zero, which can be a critical reference for understanding the behavior of a line.

In our particular case, the y-intercept is 4. By substituting \( x = 0 \) in the line equation, we confirm \( y = 4 \), showing this line intersects the y-axis at 4. It sets a clear point on the graph from which the line's slope will dictate the rise and run.

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Most popular questions from this chapter

Exercises \(51-60\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph . $$ y=x^{2}-1, \text { stretched vertically by a factor of } 3 $$

Evaluate \(\cos \frac{\pi}{12}\)

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Exercises \(51-60\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph . $$ y=\sqrt{x+1}, \quad \text { stretched vertically by a factor of } 3 $$

Exercises \(19-28\) tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ y=\frac{1}{2}(x+1)+5 \quad \text { Down } 5, \text { right } 1 $$
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