12 multiple-choice questions, progressively harder.
The graph of fff passes through (−2,5)(-2, 5)(−2,5), (−1,0)(-1, 0)(−1,0), (0,−3)(0, -3)(0,−3), (1,−4)(1, -4)(1,−4), (2,−3)(2, -3)(2,−3), (3,0)(3, 0)(3,0), and (4,5)(4, 5)(4,5). For how many of these inputs does f(x)=5f(x) = 5f(x)=5?
Solution
Correct answer: A
Count the listed inputs whose output is 555.
f(−2)=5andf(4)=5f(-2) = 5 \quad \text{and} \quad f(4) = 5f(−2)=5andf(4)=5
Two inputs share the output 555, so f(x)=5f(x) = 5f(x)=5 has two solutions among those shown.
A curve is drawn only for −2≤x≤4-2 \le x \le 4−2≤x≤4. Reading it, f(−2)=3f(-2) = 3f(−2)=3, it dips to −5-5−5 at x=1x = 1x=1, and f(4)=3f(4) = 3f(4)=3. What are its domain and range?
Correct answer: D
The domain is the horizontal spread between the endpoints, and the range is the vertical spread from the low point to the high point.
domain −2≤x≤4,range −5≤y≤3\text{domain } -2 \le x \le 4, \qquad \text{range } -5 \le y \le 3domain −2≤x≤4,range −5≤y≤3
The lowest height is −5-5−5 and the highest is 333.
The graph of ggg is a parabola with lowest point (1,−4)(1, -4)(1,−4), and it passes through (3,0)(3, 0)(3,0). By symmetry, which other point must be on the graph?
A parabola is a mirror image across the vertical line through its lowest point, here x=1x = 1x=1. The point (3,0)(3, 0)(3,0) is 222 units right of that line, so its mirror is 222 units left.
1−2=−1 ⇒ (−1,0)1 - 2 = -1 \;\Rightarrow\; (-1, 0)1−2=−1⇒(−1,0)
So (−1,0)(-1, 0)(−1,0) is on the graph.
The graph of fff meets the horizontal line y=2y = 2y=2 at three different points. What does this say about f(x)=2f(x) = 2f(x)=2?
Correct answer: C
The solutions of f(x)=2f(x) = 2f(x)=2 are the inputs where the horizontal line y=2y = 2y=2 meets the graph.
3 meeting points ⇒ 3 solutions3 \text{ meeting points} \;\Rightarrow\; 3 \text{ solutions}3 meeting points⇒3 solutions
A horizontal line may cross a graph many times; only a vertical line is limited to one crossing for a function.
A function fff has f(x)=0f(x) = 0f(x)=0 at exactly x=−3x = -3x=−3 and x=5x = 5x=5, and nowhere else. Which must be true of its graph?
The zeros of fff are exactly the x-intercepts, the points where the output is 000.
f(−3)=0 and f(5)=0 ⇒ (−3,0),(5,0)f(-3) = 0 \text{ and } f(5) = 0 \;\Rightarrow\; (-3, 0), (5, 0)f(−3)=0 and f(5)=0⇒(−3,0),(5,0)
Those are the only two crossings of the horizontal axis.
Which statement correctly distinguishes the intercepts of a graph?
Zero the coordinate of the axis you are crossing, which is the opposite letter.
y-intercept=(0,f(0)),x-intercepts solve f(x)=0\text{y-intercept} = (0, f(0)), \qquad \text{x-intercepts solve } f(x) = 0y-intercept=(0,f(0)),x-intercepts solve f(x)=0
The y-intercept sets x=0x = 0x=0; the x-intercepts set y=0y = 0y=0.
On the graph of fff, the point (a,f(a))(a, f(a))(a,f(a)) moves so that f(a)f(a)f(a) keeps growing while aaa grows. On that stretch, fff is which of these?
Larger inputs producing larger outputs is exactly the definition of increasing.
a grows and f(a) grows ⇒ increasinga \text{ grows and } f(a) \text{ grows} \;\Rightarrow\; \text{increasing}a grows and f(a) grows⇒increasing
The height of the graph climbs as you move right.
A parabola opens downward with vertex (3,5)(3, 5)(3,5). Which statement about its graph is true?
A downward parabola peaks at its vertex, giving a maximum output there.
vertex (3,5) ⇒ maximum 5 at x=3\text{vertex } (3, 5) \;\Rightarrow\; \text{maximum } 5 \text{ at } x = 3vertex (3,5)⇒maximum 5 at x=3
Its range is y≤5y \le 5y≤5, and it increases before x=3x = 3x=3 and decreases after.
A function fff satisfies f(0)=4f(0) = 4f(0)=4 and has no other point with first coordinate 000. How many times does its graph cross the y-axis?
Correct answer: B
The y-axis is the vertical line x=0x = 0x=0, and a function assigns the input 000 exactly one output.
one output at x=0 ⇒ one crossing (0,4)\text{one output at } x = 0 \;\Rightarrow\; \text{one crossing } (0, 4)one output at x=0⇒one crossing (0,4)
A graph can meet the y-axis at most once, which is why the vertical line test never fails there.
A function fff has f(x)=2f(x) = 2f(x)=2 for every input xxx. What does its graph look like?
Every input gives the same output 222, so every point has height 222.
f(x)=2 for all x ⇒ horizontal line y=2f(x) = 2 \text{ for all } x \;\Rightarrow\; \text{horizontal line } y = 2f(x)=2 for all x⇒horizontal line y=2
The outputs never change, so the graph is flat.
The point (2,b)(2, b)(2,b) lies on the graph of f(x)=x2−3x+4f(x) = x^2 - 3x + 4f(x)=x2−3x+4. What is bbb?
Since the point is on the graph, b=f(2)b = f(2)b=f(2).
f(2)=22−3(2)+4=4−6+4=2f(2) = 2^2 - 3(2) + 4 = 4 - 6 + 4 = 2f(2)=22−3(2)+4=4−6+4=2
So b=2b = 2b=2 and the point is (2,2)(2, 2)(2,2).
The graph of fff has a y-intercept at (0,−6)(0, -6)(0,−6) and x-intercepts at −2-2−2 and 333. Which of these is f(0)f(0)f(0)?
The y-intercept gives the output at input 000.
f(0)=−6f(0) = -6f(0)=−6
The values −2-2−2 and 333 are x-intercepts, where the output is 000, not f(0)f(0)f(0).
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