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Chapter 2: Problem 33

Exer. 33-36: Find the slope-intercept form of the line that satisfies the given conditions. \(x\) -intercept \(4, \quad y\) -intercept \(-3\)

Short Answer

Expert verified
The slope-intercept form is \(y = \frac{3}{4}x - 3\).

Step by step solution

01

Identify Intercepts

The problem gives us two intercepts: the x-intercept at 4, which tells us that the line crosses the x-axis at the point \((4, 0)\), and the y-intercept at -3, indicating the line crosses the y-axis at \((0, -3)\).
02

Calculate Slope

The slope \(m\) of the line can be calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the points \((4, 0)\) and \((0, -3)\), we have:\[m = \frac{-3 - 0}{0 - 4} = \frac{-3}{-4} = \frac{3}{4}\]
03

Write Slope-Intercept Form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. From the previous steps, we know the slope \(m = \frac{3}{4}\) and the y-intercept \(b = -3\). Thus, the equation of the line is:\[y = \frac{3}{4}x - 3\]

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Key Concepts

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

Understanding x-intercept
The x-intercept of a line is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. For example, in our problem, the x-intercept is given as 4. This simply means that the line touches the x-axis at the point \((4, 0)\).

To quickly locate an x-intercept on a graph, look where the line meets the horizontal axis. Knowing x-intercepts is crucial because it provides insight into where your line begins or ends as it traverses the Cartesian plane.

This point can often be found by setting \(y = 0\) in the equation of your line and solving for \(x\). This is especially useful when only the equation is given, and you need to visualize the line.
Exploring y-intercept
The y-intercept is the point where the line crosses the y-axis. At this point, the x-coordinate is zero. In our example, the y-intercept is -3, which means the line crosses the y-axis at \((0, -3)\).

Why is the y-intercept important? It shows where the line begins on the y-axis and gives the initial value of \(y\) when \(x\) is zero. This can often be the starting point in many real-world applications where the y-intercept represents the initial condition.
  • You can find the y-intercept directly from the slope-intercept form, which is \(b\) in the equation \(y = mx + b\).
Understanding the y-intercept helps in sketching the line, showcasing how the line behaves and where it crosses the y-axis.
The Slope: Calculating and Understanding
The slope of a line quantifies its steepness and direction. Given two points, the formula for calculating slope \(m\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

In our exercise, we used the points \((4, 0)\) and \((0, -3)\). Plugging these into the slope formula yields:
  • Numerator: \(-3 - 0\) from the \(y\)-coordinates.
  • Denominator: \(0 - 4\) from the \(x\)-coordinates.
  • Slope: \(\frac{-3}{-4} = \frac{3}{4}\).
This results in a positive slope, indicating the line ascends from left to right.

The slope is a vital aspect that reveals how a line changes across a graph. Knowing this helps determine whether a line rises, falls, or remains constant.
Equation of a Line in Slope-Intercept Form
The slope-intercept form of a line is an equation that provides a comprehensive view of the line's direction and starting point. It is expressed as \(y = mx + b\), where \(m\) represents the slope, and \(b\) denotes the y-intercept.

From the problem, we determined:
  • Slope \(m = \frac{3}{4}\).
  • Y-intercept \(b = -3\).
Therefore, putting these values into the slope-intercept form creates the equation \(y = \frac{3}{4}x - 3\).

This form is quite handy, not just for graphing, but also as a practical method for prediction and analysis in various fields, such as economics and physics, where linear relationships are analyzed. Understanding this form aids in constructing graphs efficiently and in interpreting linear models.

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Most popular questions from this chapter

Precipitation in Seattle The average monthly precipitation (in inches) for Seattle is listed in the following table. (Note: April average is not given.) (a) Plot the average monthly precipitation. (b) Model the data with a quadratic function of the form \(f(x)=a(x-h)^{2}+k .\) Graph \(f\) and the data on the same coordinate axes. (c) Use \(f\) to predict the average rainfall in April. Compare your prediction with the actual value of 2.55 inches. $$\begin{array}{|l|c|} \hline \text { Month } & \text { Precipitation } \\ \hline \text { Jan. } & 5.79 \\ \hline \text { Feb. } & 4.02 \\ \hline \text { Mar. } & 3.71 \\ \hline \text { April } & \\ \hline \text { May } & 1.70 \\ \hline \text { June } & 1.46 \\ \hline \text { July } & 0.77 \\ \hline \text { Aug. } & 1.10 \\ \hline \text { Sept. } & 1.72 \\ \hline \text { Oct. } & 3.50 \\ \hline \text { Nov. } & 5.97 \\ \hline \text { Dec. } & 5.81 \\ \hline \end{array}$$

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