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Chapter 3: Problem 26

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ y=\frac{1}{4} x-1 $$

Short Answer

Expert verified
The y-intercept is (0, -1) and the x-intercept is (4, 0).

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( y \). y = \( \frac{1}{4}(0) - 1 \)= -1So, the y-intercept is \( (0, -1) \).
02

- Find the x-intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \).0 = \( \frac{1}{4}x - 1 \)Add 1 to both sides to isolate the term with \(x\):1 = \( \frac{1}{4}x \)Multiply both sides by 4 to solve for \( x \):x = 4So, the x-intercept is \( (4, 0) \).

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Key Concepts

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

Understanding the y-Intercept
The y-intercept is where the graph of an equation crosses the y-axis. This point always has an x-coordinate of zero. To find the y-intercept, you need to set x to zero in the equation and solve for y.
If you have the equation: \( y = \frac{1}{4}x - 1 \), you set \( x = 0 \) to get:
\( y = \frac{1}{4}(0) - 1 \)
Simplifying this, we find that:
\( y = -1 \).
This means the y-intercept is the point \( (0, -1) \).
Remember, for any linear equation of the form \( y = mx + b \), the y-intercept is always \( (0, b) \).
Finding the x-Intercept
The x-intercept is where the graph crosses the x-axis. At this point, the y-coordinate is zero. To find the x-intercept, set y to zero in the equation and solve for x.
In the equation: \( y = \frac{1}{4}x - 1 \), setting \( y = 0 \) gives:
\( 0 = \frac{1}{4}x - 1 \)
Add 1 to both sides:
\( 1 = \frac{1}{4}x \)
Now, multiply both sides by 4 to solve for x:
\( x = 4 \)
So, the x-intercept is the point \( (4, 0) \).
It鈥檚 important to remember that finding intercepts involves making one of the variables zero and solving for the other.
Linear Equations and Their Graphs
A linear equation is an equation that forms a straight line when graphed. Its general form is \( y = mx + b \), where:
  • \(m\) is the slope, which tells you the steepness of the line
  • \(b\) is the y-intercept, or where the line crosses the y-axis
For example, the equation \( y = \frac{1}{4}x - 1 \) has a slope of \( \frac{1}{4} \) and a y-intercept of -1. When graphed, this line will be less steep than a line with a greater slope and will cross the y-axis at -1.

Finding the intercepts not only helps in graphing the line but also gives you a clear idea of where the line crosses the axes, which is crucial for understanding the behavior of the line in a coordinate plane.

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Most popular questions from this chapter

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Passes through (4,2) perpendicular to \(x-3 y=7\)

Solve each problem. Suppose that it costs a flat fee of \(\$ 20\) plus \(\$ 15\) per day to rent a pressure washer. Therefore, the cost \(y\) in dollars to rent the pressure washer for \(x\) days is given by the linear equation $$ y=15 x+20 $$ Express each of the following as an ordered pair. (a) When the washer is rented for 5 days, the cost is \(\$ 95\). (b) When the cost is \(\$ 110\), the washer is rented for 6 days.

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ 3 x+y=0 $$

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Parallel to \(3 x+y=7\) \(y\) -intercept (0,4)

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Passes through (2,3) parallel to \(4 x-y=-2\)
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