4.1E: Linear Functions (Exercises)
- Page ID
- 56076
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. Determine whether the algebraic equation is linear. \(2 x+3 y=7\)
2. Determine whether the algebraic equation is linear. \(6 x^{2}-y=5\)
3. Determine whether the function is increasing or decreasing. \[f(x)=7 x-2 \nonumber\]
4. Determine whether the function is increasing or decreasing. \[g(x)=-x+2 \nonumber\]
5. Given each set of information, find a linear equation that satisfies the given conditions, if possible.
Passes through (7,5) and (3,17)
6. Given each set of information, find a linear equation that satisfies the given conditions, if possible.
\(x\) -intercept at (6,0) and \(y\) -intercept at (0,10)
7. Find the slope of the line shown in the graph.
8
9.
10
Does the following table represent a linear function? If so, find the linear equation that models the data.
| x | –4 | 0 | 2 | 10 |
|---|---|---|---|---|
| g(x) | 18 | –2 | –12 | –52 |
Does the following table represent a linear function? If so, find the linear equation that models the data.
| x | 6 | 8 | 12 | 26 |
|---|---|---|---|---|
| g(x) | –8 | –12 | –18 | –46 |
12. On June \(1^{\text {st }}\), a company has \(\$ 4,000,000\) profit. If the company then loses 150,000 dollars per day thereafter in the month of June, what is the company's profit \(n^{\text {th }}\) day after June \(1^{\text {st }}\) ?
For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
13. \[\begin{align*} 2 x-6 y &=12 \\[4pt] -x+3 y &=1 \end{align*}\]
14 \[\begin{align*} y &=\frac{1}{3} x-2 \\[4pt] 3 x+y &=-9 \end{align*}\]
For the following exercises, find the \(x\) - and \(y\) - intercepts of the given equation
15. \(7 x+9 y=-63\)
16. \(f(x)=2 x-1\)
For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line \(2 .\) Is each pair of lines parallel, perpendicular, or neither?
17.
- Line 1 : Passes through (5,11) and (10,1)
- Line 2: Passes through (-1,3) and (-5,11)
18.
- Line 1: Passes through (8,-10) and (0,-26)
- Line 2: Passes through (2,5) and (4,4)
19. Write an equation for a line perpendicular to \(f(x)=5 x-1\) and passing through the point (5,20).
20. Find the equation of a line with a \(y\) - intercept of (0,2) and slope \(-\frac{1}{2}\).
23. A car rental company offers two plans for renting a car.
- Plan A: 25 dollars per day and 10 cents per mile
- Plan B: 50 dollars per day with free unlimited mileage
How many miles would you need to drive for plan B to save you money?