➕ Algebra 1 · Quadratics

Graphing Quadratic Functions practice questions

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Answer key with explanations

1. Which way does the graph of y = −4x² + x + 5 open?

   Answer: Downward, because a = −4 is negative. Only the sign of a decides the direction, and here a = −4, so the parabola opens down. The positive b and c do not matter. The fact that x² is never negative is true, but x² is multiplied by −4. And an x term is allowed: y = ax² + bx + c has one.

2. What is the axis of symmetry of y = x² − 12x + 7?

   Answer: x = 6. Here a = 1 and b = −12, so x = −b/(2a) = −(−12)/(2 · 1) = 12/2 = 6. The choice x = −6 forgets to flip the sign of b. The choice x = 12 forgets to divide by 2. The choice x = −12 makes both mistakes. Check: the points at x = 5 and x = 7 are each 1 from the axis, and both have y = −28.

3. At which point does the graph of y = 3x² − 7x + 5 cross the y-axis?

   Answer: (0, 5). Every point on the y-axis has x = 0. Then y = 3(0)² − 7(0) + 5 = 5, so the graph crosses at (0, 5). The y-intercept is c = 5. The number −7 is b, and 3 is a. The point (5, 0) has its coordinates swapped, and it is not even on the graph: at x = 5 the height is 75 − 35 + 5 = 45.

4. What is the vertex of y = 2x² + 12x + 7?

   Answer: (−3, −11). Here a = 2 and b = 12, so x = −12/(2 · 2) = −12/4 = −3. Then y = 2(−3)² + 12(−3) + 7 = 18 − 36 + 7 = −11. The vertex is (−3, −11). The point (3, 61) comes from forgetting to flip the sign of b: x = 3 gives 18 + 36 + 7 = 61. The point (−6, 7) divides by a instead of 2a: x = −6 gives 72 − 72 + 7 = 7. The point (−3, 7) uses the right x but the y-intercept 7 instead of substituting.

5. The height of a tossed ball is modeled by h = −16t² + 32t + 4, where h is in feet and t is in seconds after the toss. In this model, what is the ball’s greatest height?

   Answer: 20 feet. Since a = −16 is negative, the vertex is the highest point. The axis is t = −32/(2 · (−16)) = 1, and h = −16(1)² + 32(1) + 4 = 20. The greatest height is 20 feet, reached at 1 second. The choice 1 is the time, not a height. The choice 4 is the starting height. The choice 36 comes from adding 32 + 4 and ignoring the −16t² term.

6. A parabola has axis of symmetry x = 10 and passes through (4, 9). Which other point must be on the parabola?

   Answer: (16, 9). The x-value 4 is 6 units left of the axis x = 10, so the mirror point is 6 units right of it: x = 10 + 6 = 16, with the same y-value, 9. The point (−4, 9) reflects across the y-axis instead of the axis x = 10. The point (14, 9) is 4 units right of the axis, but the distance should be 6. The point (16, −9) flips the y-value, but a mirror point across a vertical axis keeps the same height.

7. The graph of y = (x + 4)(x − 10) crosses the x-axis at x = −4 and x = 10. What is its axis of symmetry?

   Answer: x = 3. The axis is halfway between the x-intercepts: (−4 + 10)/2 = 6/2 = 3. Check with the formula: (x + 4)(x − 10) = x² − 6x − 40, so −b/(2a) = 6/2 = 3. The choice x = −3 has the wrong sign. The choice x = 6 adds the zeros but forgets to divide by 2. The choice x = 7 averages 4 and 10 and ignores that the first zero is −4.

8. A parabola opens downward and its vertex is (−6, −4). How many x-intercepts does it have?

   Answer: None: the vertex is the highest point, and it is already below the x-axis. A parabola that opens down is at its highest at the vertex, so every y-value is at most −4. That is below the x-axis, so no point has y = 0 and the parabola has no x-intercepts. Width does not change this: any a < 0 gives the same answer. A parabola can have two, one or no x-intercepts, so it does not always cross twice. It touches the axis at its vertex only if the vertex is on the axis, which would need a y-value of 0.

9. Which equation has a graph that opens downward, crosses the y-axis at (0, 5), and has axis of symmetry x = 1?

   Answer: y = −x² + 2x + 5. For y = −x² + 2x + 5: a = −1 < 0 opens down, c = 5 gives the point (0, 5), and −b/(2a) = −2/(−2) = 1. Each other choice fails one test. y = x² − 2x + 5 has a = 1 and opens up. y = −x² − 2x + 5 has axis −(−2)/(2 · (−1)) = −1. And y = −x² + 2x − 5 has c = −5, so it crosses the y-axis at (0, −5).

10. A rectangle has a perimeter of 36 feet. If its width is w feet, its length is 18 − w feet, so its area in square feet is A = w(18 − w) = −w² + 18w. What is the largest possible area?

    Answer: 81 square feet. Since a = −1 is negative, the vertex is the highest point. The axis is w = −18/(2 · (−1)) = 9, and A = −9² + 18(9) = −81 + 162 = 81. The largest area is 81 square feet, when the width is 9 feet and the length is 18 − 9 = 9 feet. The choice 9 is the width, not the area. The choice 18 is the number b. The choice 162 is only the 18w part, without subtracting w².