Math assignment 1

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Assignment #1

MAC 1140 – Spring 2020 – Due: March 27, 2020

1. Determine whether each of the following relationships is a function and give the domain and range for each relation.

Relation Function?

Yes No Domain Range

(a) {(1,4), (4,4), (1,3), (6,3)}

(b) {(1,3), (2,5), (5,6), (3,7)}

(c) {(−3, −3), (−1, −1), (7,0)}

2. Determine whether each of the following equations defines 𝑦 as a function of 𝑥.

Equation Function?

Explain your rationale Yes No

(a) 𝑥 + 𝑦 = −3

(b) 𝑥 + 𝑦2 = −3

(c) |𝑥| + 𝑦 = 0

(d) 𝑥2 + 𝑦2 = 1

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3. You are given the following function. Find: (a) The domain of the function: ___________

(b) The range of the function: ___________

(c) The 𝑥-intercepts (approximate, as necessary): _____________________

(d) The 𝑦-intercept: ___________

(e) Intervals on which 𝑓(𝑥) is decreasing: _________________________________

(f) Intervals on which 𝑓(𝑥) is increasing: _________________________________

(g) Intervals on which 𝑓(𝑥) is constant (if any): ____________________________

(h) The values of 𝑥 where 𝑓(𝑥) = 3 (approximate, as necessary): _______________

(i) The relative maxima of the function: (𝑥, 𝑦) = ____________________________

(j) The relative minima of the function: (𝑥, 𝑦) = ____________________________

(k) The absolute maximum of the function: (𝑥, 𝑦) = ________

(l) The absolute minimum of the function: (𝑥, 𝑦) = ________

(m) Is the function even odd, or neither? ______________

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4. Sketch the graph of 𝑓(𝑥) = 𝑥2 − 4𝑥 and identify all the relative extrema and absolute extrema of the function on each of the following intervals.

(a) (−∞, +∞) (b) [−1,4] (c) [1,3] (d) [3,5] (e) (−1,5]

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5. Find the difference quotient for the following functions: (a) 𝑓(𝑥) = 3

(b) 𝑓(𝑥) = −2𝑥

(c) 𝑓(𝑥) = 4𝑥2

(d) 𝑓(𝑥) = 5𝑥2 + 2𝑥 − 4

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6. For each case below, use the conditions to write an equation for each line in point-slope form and slope-intercept form. (a) slope = −2, passing through (3,5) (b) passing through (0, −2), with 𝑥-intercept = −1 (c) passing through (0, −2) and (3,5) (d) 𝑥-intercept = 2, with 𝑦-intercept = −3 (e) passing through (1,1), parallel to the line 𝑦 = −3𝑥 + 2

(f) passing through (0,0), perpendicular to the line 𝑦 = 1

3 𝑥 − 2

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7. (a) Find the distance between the points (2, −3) and (5,4). If necessary, express your answer in simplified radical form and then round to two decimal points. (b) Find the midpoint of the line segment with endpoints at (2, −3) and (5,4). (c) Write the standard form of the equation of the circle with radius 𝑟 = 2 and center (−1,3). (d) Find the center and radius of the circle given by 𝑥2 + 𝑦2 − 4𝑥 + 2𝑦 − 4 = 0.

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8. You are given the quadratic function 𝑓(𝑥) = −𝑥2 − 2𝑥 + 3. (a) Without graphing, determine if the function has a minimum or maximum and explain your

reasoning. (b) The coordinates of the vertex are: (c) The intercepts are: (d) The equation of the parabola’s axis of symmetry is: (e) Graph the function. (f) The domain of the function is: (g) The range of the function is: