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Chapter 0: Problem 52

Find the \(x\) - and \(y\)-intercepts of the graph of the equation. \(y=8-3 x\)

Short Answer

Expert verified
The x-intercept is \(x = 8/3\) and the y-intercept is \(y = 8\).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\) in the equation \(0 = 8 - 3x\). Solving for \(x\) we find \(x = 8/3\).
02

Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\) in the equation \(y = 8 - 3(0)\). Solving for \(y\) we find \(y = 8\)

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Key Concepts

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

X-Intercept
Understanding the x-intercept of a graph is crucial when analyzing linear equations. The x-intercept occurs where the graph of the equation crosses the x-axis. At this point, the value of y is zero. To find the x-intercept, you simply set the y-coordinate to zero and solve the equation for x.

For the equation given in the exercise, \( y=8-3x \), you can find the x-intercept by substituting y with 0 which gives us \( 0=8-3x \). Upon solving this, \( x=\frac{8}{3} \), which is approximately 2.67. This means the graph crosses the x-axis at this point.
Y-Intercept
The y-intercept is found where the equation's graph intersects the y-axis, with the x-coordinate being zero at that point. To calculate the y-intercept for \( y=8-3x \) as in our exercise, we plug in 0 for x.

Thus, \( y=8-3(0) \). This simplifies to \( y=8 \), showing that the graph intersects the y-axis at the point (0, 8). It is helpful to remember that the y-intercept represents the point where the function has its initial value when no changes have been made to the independent variable, here being x.
Linear Equations
Linear equations form straight lines when graphed on a coordinate plane. They have the general form \( y=mx+b \), where \(m\) is the slope of the line and \(b\) is the y-intercept. Our exercise features a linear equation with a negative slope, which means the graph will slant downwards as it moves from left to right.

Knowing how to work with linear equations is fundamental in mathematics as they are used to represent numerous real-world situations, such as calculating rates or predicting trends. Solving linear equations for their intercepts provides insights into the behavior of the line, such as where it starts and in which direction it goes.
Trigonometry
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. Although our current exercise is focused on linear equations, understanding trigonometry is essential for a more comprehensive exploration of graphs, especially when dealing with periodic functions like sine and cosine.

These functions graphically depict waves and have x-intercepts and y-intercepts that can tell us about the angles and side lengths in trigonometric contexts. While trigonometry is not directly applicable to this linear equation, it's a powerful tool when moving beyond linear relationships into more complex graph analyses.

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Most popular questions from this chapter

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=-2 x+15 & x_{1}=0, x_{2}=3 \end{array} $$

College Students The numbers of foreign students \(F\) (in thousands) enrolled in colleges in the United States from 1992 to 2002 can be approximated by the model. $$ F=0.004 t^{4}+0.46 t^{2}+431.6, \quad 2 \leq t \leq 12 $$ where \(t\) represents the year, with \(t=2\) corresponding to 1992. (Source: Institute of International Education) (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 1992 to 2002. Interpret your answer in the context of the problem. (c) Find the five-year time periods when the rate of change was the greatest and the least.

Think About It Consider \(f(x)=\sqrt{x-2}\) and \(g(x)=\sqrt[3]{x-2}\). Why are the domains of \(f\) and \(g\) different?

True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. If the inverse function of \(f\) exists and the graph of \(f\) has a \(y\)-intercept, the \(y\)-intercept of \(f\) is an \(x\)-intercept of \(f^{-1}\).

Each function models the specified data for the years 1995 through 2005 , with \(t=5\) corresponding to 1995 . Estimate a reasonable scale for the vertical axis (e.g., hundreds, thousands, millions, etc.) of the graph and justify your answer. (There are many correct answers.) (a) \(f(t)\) represents the average salary of college professors. (b) \(f(t)\) represents the U.S. population. (c) \(f(t)\) represents the percent of the civilian work force that is unemployed.
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