/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 Graph each equation in Exercises... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 28

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=-\frac{1}{2} x+2\)

Short Answer

Expert verified
The graph of the equation \(y=-\frac{1}{2}x+2\) is a straight line that descends from left to right. The line passes through the points corresponding to integer values of \(x\) from \(-3\) to 3 inclusive, with a y-intercept at (0,2).

Step by step solution

01

Identify the slope and y-intercept

The given equation is \(y=-\frac{1}{2}x+2\). In this equation, the slope \(m\) is \(-\frac{1}{2}\) and the y-intercept \(b\) is 2.
02

Plot the y-intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. Hence, plot a point at (0,2).
03

Plot other points using the slope

The slope of the line, \(-\frac{1}{2}\), tells us that for every step we go to the right on the x-axis, we go down a half step on the y-axis. So, from the point (0,2), move to the right and downward to plot more points. The chosen integer values for \(x\) range from \(-3\) to 3. Substitute these values into the equation to find the corresponding \(y\) values. For example, if \(x=1\), then \(y=-\frac{1}{2}*1+2 = 1.5\). Hence, plot a point at (1,1.5). Repeat this process for all integer values of \(x\) from \(-3\) to 3.
04

Draw the line

Once all the points are plotted on the graph, draw a straight line that passes through all these points. This line is the graph of the equation \(y=-\frac{1}{2}x+2\).

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Most popular questions from this chapter

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \boldsymbol{x}, \text { Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y}, \\ \text { Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 0.5 & 7.4 \\ \hline 1.5 & 9 \\ \hline 4 & 6 \\ \hline \end{array} $$ $$ \begin{array}{|l|} \hline u a d R e g \\ y=a \times 2+b x+c \\ a=-.8 \\ b=3.2 \\ c=6 \end{array} $$ a. Explain why a quadratic function was used to model the data. Why is the value of \(a\) negative? b. Use the graphing calculator screen to express the model in function notation. c. Use the model from part (b) to determine the \(x\)-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the ball occurs _____ feet from where it was thrown, and the maximum height is _____ feet.

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{r}2 x+y<3 \\ x-y>2\end{array}\right.\)

In Exercises 39-40, write each sentence as an inequality in two variables. Then graph the inequality. The \(y\)-variable is at least 4 more than the product of \(-2\) and the \(x\)-variable.

Describe the shape of a scatter plot that suggests modeling the data with a quadratic function.

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \boldsymbol{x} \text {, Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y}, \\ \text { Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 1 & 7.6 \\ \hline 3 & 6 \\ \hline 4 & 2.8 \\ \hline \end{array} $$ $$ \begin{aligned} &\text { [adadReg }\\\ &\begin{aligned} &y=3 \times 2+b x+c \\ &\bar{y}=.8 \\ &b=2.4 \\ &c=6 \end{aligned} \end{aligned} $$ a. Explain why a quadratic function was used to model the data. Why is the value of \(a\) negative? b. Use the graphing calculator screen to express the model in function notation. c. Use the model from part (b) to determine the \(x\)-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the ball occurs ____ feet from where it was thrown and the maximum height is ____ feet.
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