Calculus Graphs Practice Questions

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Calculus Graphs Practice Questions

Plot the following points on a Cartesian plane.

(3,5)
(–3,4)
(2, –6)
(–1, – 5)
(0,3)
(– 5,0)
(0,0)

For the function f(x) = x² – 2x + 5, plot the point at the following positions.

x = 3
x = 1
x = 0
x = –2

Use the graph of each function to determine the domain, the discontinuities, where the function is increasing and decreasing, the local maximum and minimum points, where the function is concave up and down, the points of inflection, and the asymptotes.

Find the slope between the following points.

(1,5) and (2,8)
(2,5) and (6,7)
(7,3) and (–2,3)
(–2, –4) and (–6, 5)
(2,7) and (5,w)
(4,10) and (x,y)

Find the equation of the straight line with the given information and then graph the line.

slope 2 through point (1,–2)
slope through point (6,1)
through points (5,3) and (–1, – 3)
through points (2,5) and (6,5)

Solutions

Graphs

The domain of f is x ≠ 0. There is a discontinuity at x = 0. The graph of f is decreasing on (–∞,0) and on (0,∞). The graph of f is concave down on (–∞,0) and concave up on (0,∞). There are no points of inflection, no local maxima, and no local minima. There is a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.
The domain of g consists of all real numbers. There are no discontinuities. The function increases on (–∞,–3) and on (0,3), and it decreases on (–3,0) and on (3,∞). There are local maxima at (–3,4) and (3,4), and there is a local minimum at (2,0). The graph is concave up on (–1,1) and concave down on (–∞, –1) and on (1,∞). There are points of inflection at (∞1,3) and (1,3). There are no asymptotes.
The domain of h is x ≠ 1. There is a discontinuity at x = 1. The function increases on (–1,1) and on (1,∞), and decreases on (–∞,–1). The function is concave up on (–∞,1) and on (1,∞). There is a local minimum at (1,–2). There are no asymptotes, nor any points of inflection.
The domain is x ≠ – 2,2 with discontinuities at x = – 2 and x = 2. The function increases on (0,2) and (2,∞), and decreases on (–∞, – 2) and on (–2,0). The point (0,2) is a local minimum. The graph is concave up on (–2,2) and concave down on (–∞, – 2) and (2,∞). There are no points of inflection. There are vertical asymptotes at x = –2 and x = 2, and a horizontal asymptote at y = 0.
The domain consists of all real numbers, though there is a discontinuity at x = –1. The function increases on (–∞, –1) and on (–1,2), and it decreases on (2,∞). There are local maxima at (–1,3) and (2,3). The graph is concave up on (–1,0) and concave down on (0,∞), so there is a point of inflection at (0,2). Because the line is straight before x = –1, it does not curve upward or downward, and thus has no concavity. There are no asymptotes.
The domain is the whole real line, with no discontinuities. The graph increases on ( –∞,∞), is concave up on ( –∞,0), and is concave down on (0,∞). There is a point of inflection at (0,0). There are horizontal asymptotes at y = –2 and y= 2.
The domain is (0,∞) with no discontinuities. The graph increases on (0,2), has a local maximum at (2,5), and decreases on (2,∞). The graph is concave down on (0,3) and concave up on (3,∞) with a point of inflection at (3,3). There is a vertical asymptote at x = 0 and a horizontal asymptote at y = 1.
The domain is x ≠5 with discontinuities at x = 2 and x = 5. The function increases on (-∞,–1),(4,5), and on (5,∞). The function decreases on (1,2) and on (2,4). There is a local maximum at (1,2) and at (2,3). The point (4,2) is a local minimum. The graph is concave up on (– ∞,1), (1,2), (2,5), and on (5,∞). There is a horizontal asymptote at y = 0.
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