Quadratic Functions and Graphs: math practice, Grade 9 – download the PDF

1 Reading a, b, and c ★★★

Write each function in standard form if needed, then give \(a\), \(b\), and \(c\).

1. \(y = 3x^2 - 5x + 2\)
2. \(y = -x^2 + 4\)
3. \(y = 0.5x^2 + x - 7\)
4. \(y = 6 - 2x - x^2\)

2 Up or down, and the y-intercept ★★★

For each function, say whether the parabola opens up or down and give the \(y\)-intercept.

1. \(y = 2x^2 - 9\)
2. \(y = -0.5x^2 + 3x + 1\)
3. \(y = -4x^2 + x\)
4. \(y = x^2 + 8x + 3\)

3 A first vertex ★★★

Find the axis of symmetry and the vertex of \(y = x^2 - 4x + 1\). Is the vertex a maximum or a minimum?

4 Completing a table ★★★

Complete the table for \(y = x^2 - 2x - 3\), then give the vertex and the axis of symmetry.

5 Reading vertex form ★★★

Give the vertex of each parabola and say whether it opens up or down.

1. \(y = (x - 4)^2 + 2\)
2. \(y = -3(x + 1)^2 - 5\)
3. \(y = \dfrac{1}{2}(x - 6)^2\)
4. \(y = (x + 7)^2 + 9\)

6 Describing transformations ★★★

Describe how each graph is related to the graph of \(y = x^2\).

1. \(y = x^2 - 5\)
2. \(y = (x + 3)^2\)
3. \(y = -x^2\)
4. \(y = 2x^2\)
5. \(y = (x - 1)^2 + 4\)

7 Zeros from factored form ★★★

Find the zeros of each function.

1. \(y = (x - 2)(x + 6)\)
2. \(y = x(x - 9)\)
3. \(y = (2x - 5)(x + 1)\)

8 Graphing from the vertex ★★★

Graph \(y = x^2 + 2x - 8\). Find the axis of symmetry, the vertex, the \(y\)-intercept and its mirror point, and the \(x\)-intercepts.

9 Standard form from vertex form ★★★

Write \(y = 2(x - 3)^2 - 5\) in standard form \(ax^2 + bx + c\).

10 Equation from a vertex and a point ★★★

A parabola has its vertex at \((-2, 5)\) and passes through \((0, 1)\). Write its equation in vertex form.

11 Solving by factoring ★★★

Solve \(x^2 - 3x - 28 = 0\) by factoring, and check both answers.

12 True or false? ★★★

Decide whether each statement is true or false and justify your answer.

1. The zeros of \(y = 3x^2 - 12\) are 2 and \(-2\).
2. The graph of \(y = x^2 + 4\) has two \(x\)-intercepts.
3. The axis of symmetry of \(y = x^2 - 10x + 3\) is \(x = 5\).

13 Maximum and minimum values ★★★

Find the maximum or minimum value of each function and the \(x\)-value where it occurs.

1. \(y = x^2 - 8x + 19\)
2. \(y = -2x^2 + 12x - 5\)

14 The dog run ★★★

A rectangular dog run is built with 60 feet of fencing used on all four sides. Let \(w\) be the width in feet.

1. Write the length in terms of \(w\) and then the area \(A(w)\).
2. What width gives the largest area, and what is that area?

15 Using the quadratic formula ★★★

Solve \(2x^2 - 5x - 3 = 0\) with the quadratic formula. Compute the discriminant first.

16 A thrown ball ★★★

A ball is thrown upward from a roof. Its height in feet after \(t\) seconds is \(h(t) = -16t^2 + 32t + 6\). Use a calculator for decimals.

1. How high is the roof?
2. Find the maximum height and when it happens. Convert it to meters (1 ft = 0.3048 m).
3. When does the ball hit the ground? Round to the nearest hundredth.

17 Smoothie revenue ★★★

A snack stand finds that at a price of \(p\) dollars it sells \(90 - 5p\) smoothies per day.

1. Write the daily revenue \(R(p)\) as a quadratic function in standard form.
2. Which price gives the greatest revenue, and what is it?
3. Explain why the revenue is 0 at \(p = 18\).

18 Equation from the zeros ★★★

A parabola has zeros \(-1\) and \(5\), and it passes through the point \((2, -18)\).

1. Write its equation in factored form \(y = a(x + 1)(x - 5)\) and find \(a\).
2. Write it in standard form.
3. What is the vertex?

19 A full description ★★★

For \(y = -\dfrac{1}{2}(x + 4)^2 + 6\), give the vertex, the axis of symmetry, the transformations of \(y = x^2\), the \(y\)-intercept and its mirror point, and the range.

20 A parabolic arch ★★★

A tunnel arch over a road 12 meters wide has the shape \(y = -0.25x^2 + 3x\), where \(x\) and \(y\) are in meters and the road runs from \(x = 0\) to \(x = 12\).

1. How tall is the arch at its highest point?
2. A truck 3 meters wide and 8 meters tall drives down the middle of the road. Does it fit?

21 Counting solutions with k ★★★

Consider \(x^2 - 6x + k = 0\).

1. For which value of \(k\) is there exactly one solution?
2. How many solutions are there when \(k = 5\)? When \(k = 12\)?