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Mathematics LibreTexts

2.2E: Graphs of Linear Functions (Exercises)

  • Page ID
    13898
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    Section 2.2 exercise

    Match each linear equation with its graph

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    1. \(f(x)=-x-1\)

    2. \(f(x)=-2x-1\)

    3. \(f(x)=-\frac{1}{2} x-1\)

    4. \(f(x)=2\)

    5. \(f(x)=2+x\)

    6. \(f(x)=3x+2\)

    Sketch a line with the given features

    7. An x-intercept of (-4, 0) and y-intercept of (0, -2)

    8. An x-intercept of (-2, 0) and y-intercept of (0, 4)

    9. A vertical intercept of (0, 7) and slope \(-\dfrac{3}{2}\)

    10. A vertical intercept of (0, 3) and slope \(\dfrac{2}{5}\)

    11. Passing through the points (-6,-2) and (6,-6)

    12. Passing through the points (-3,-4) and (3,0)

    Sketch the graph of each equation

    13. \(f(x)=-2x-1\)

    14. \(g(x)=-3x+2\)

    15. \(h(x)=\dfrac{1}{3} x+2\)

    16. \(k(x)=\dfrac{2}{3} x-3\)

    17. \(k(t)=3+2t\)

    18. \(p(t)=-2+3t\)

    19. \(x=3\)

    20. \(x=-2\)

    21. \(r(x)=4\)

    22. \(q(x)=3\)

    23. If \(g(x)\) is the transformation of \(f(x)=x\) after a vertical compression by \(3/4\), a shift left by 2, and a shift down by 4

    a. Write an equation for \(g(x)\)
    b. What is the slope of this line?
    c. Find the vertical intercept of this line.

    24. If \(g(x)\) is the transformation of \(f(x)=x\) after a vertical compression by \(1/3\), a shift right by 1, and a shift up by 3

    a. Write an equation for \(g\left(x\right)\)
    b. What is the slope of this line?
    c. Find the vertical intercept of this line.

    Write the equation of the line shown

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    Find the horizontal and vertical intercepts of each equation

    29. \(f(x)=-x+2\)

    30. \(g(x)=2x+4\)

    31. \(h(x)=3x-5\)

    32. \(k(x)=-5x+1\)

    33. \(-2x+5y=20\)

    34. \(7x+2y=56\)

    Given below are descriptions of two lines. Find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular or neither?

    35. Line 1: Passes through \((0,6)\) and \((3,-24)\)
    Line 2: Passes through \((-1,19)\) and \((8,-71)\)

    36. Line 1: Passes through \((-8, -55)\) and \((10, 89)\)
    Line 2: Passes through \((9, -44)\) and \((4,-14)\)

    37. Line 1: Passes through \((2,3)\) and \((4,-1)\)
    Line 2: Passes through \((6,3)\) and \((8,5)\)

    38. Line 1: Passes through \((1, 7)\) and \((5,5)\)
    Line 2: Passes through \((-1,-3)\) and \((1,1)\)

    39. Line 1: Passes through \((0, 5)\) and \((3,3)\)
    Line 2: Passes through \((1,-5)\) and \((3,-2)\)

    40. Line 1: Passes through \((2,5)\) and \((5,-1)\)
    Line 2: Passes through \((-3,7)\) and \((3,-5)\)

    41. Write an equation for a line parallel to \(f(x)=-5x-3\) and passing through the point (2,-12)

    42. Write an equation for a line parallel to \(g(x)=3x-1\) and passing through the point (4,9)

    43. Write an equation for a line perpendicular to \(h(t)=-2t+4\) and passing through the point (-4,-1)

    44. Write an equation for a line perpendicular to \(p(t)=3t+4\) and passing through the point (3,1)

    45. Find the point at which the line \(f(x)=-2x-1\) intersects the line \(g(x)=-x\)

    46. Find the point at which the line \(f(x)=2x+5\) intersects the line \(g(x)=-3x-5\)

    47. Use algebra to find the point at which the line \(f(x)= -\dfrac{4}{5} x +\dfrac{274}{25}\) intersects the line \(h(x)=\dfrac{9}{4} x +\dfrac{73}{10}\)

    48. Use algebra to find the point at which the line \(f(x)=\dfrac{7}{4} x +\dfrac{457}{60}\) intersects the line \(g(x)=\dfrac{4}{3} x +\dfrac{31}{5}\)

    49. A car rental company offers two plans for renting a car.
    Plan A: 30 dollars per day and 18 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?

    50. You’re comparing two cell phone companies.
    Company A: $20/month for unlimited talk and text, and $10/GB for data.
    Company B: $65/month for unlimited talk, text, and data.
    Under what circumstances will company A save you money?

    Find a formula for each piecewise defined function.

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    53. Sketch an accurate picture of the line having equation \(f(x)=2-\dfrac{1}{2} x\). Let \(c\) be an unknown constant. [UW]

    a. Find the point of intersection between the line you have graphed and the line \(g(x)=1+cx\); your answer will be a point in the \(xy\) plane whose coordinates involve the unknown \(c\).
    b. Find \(c\) so that the intersection point in (a) has \(x\)-coordinate 10.
    c. Find \(c\) so that the intersection point in (a) lies on the \(x\)-axis.

    Answer

    1. E

    3. D

    5. B

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    23. a. \(g(x) = \dfrac{3}{4} (x + 2) - 4\)
    b. 3/4
    c. -5/2

    25. \(y = 3\)

    27. \(x = -3\)

      Vertical Intercept Horizontal Intercept
    29. (0, 2) (2, 0)
    31. (0, -5) (5/3, 0)
    33. (0, 4) (-10, 0)

    35. Line 1: \(m = -10\) Line 2: \(m = -10\) Parallel

    37. Line 1: \(m = -2\) Line 2: \(m = 1\) Neither

    39. Line 1: \(m = -\dfrac{2}{3}\) Line 2: \(m = \dfrac{3}{2}\) Perpendicular

    41. \(y = -5x - 2\)

    43. \(y = \dfrac{1}{2} t + 1\)

    45. (-1, 1)

    47. (1.2, 10)

    49. Plan B saves money if the miles are > 111\(\dfrac{1}{9}\)

    51. \(f(x) = \begin{cases} 2x + 3 & if & -3 \le x < -1 \\ x - 1 & if & -1\le x \le 2 \\ -2 & if & 2 < x \le 5 \end{cases}\)