Quadratic Functions and Equations: math practice, Grade 11 – download the PDF

1 Reading the coefficients ★★★

Consider \(f(x) = -3x^2 + 12x - 2\).

1. Give \(a\), \(b\) and \(c\). Does the parabola open up or down?
2. Find the axis of symmetry and the vertex.
3. Give the y-intercept.

2 Reading the vertex form ★★★

A function is given by \(g(x) = 2(x + 4)^2 - 9\). Find its vertex, its axis of symmetry, whether it has a maximum or a minimum, and its y-intercept.

3 Factoring simple trinomials ★★★

Factor each expression, then solve \(x^2 + 9x + 20 = 0\).

1. \(x^2 + 9x + 20\)
2. \(x^2 - x - 12\)
3. \(x^2 - 64\)

4 Common factor ★★★

Solve \(5x^2 - 35x = 0\) by factoring.

5 Square root method ★★★

Solve \(3(x - 2)^2 = 48\).

6 Using the discriminant ★★★

Without solving, tell how many real solutions each equation has.

1. \(2x^2 - 3x + 5 = 0\)
2. \(x^2 + 6x + 9 = 0\)
3. \(2x^2 + 7x - 4 = 0\)

7 Reading a graph for an inequality ★★★

The graph shows \(f(x) = (x + 1)(x - 4)\).

1. Where does the graph cross the x-axis?
2. Solve \(f(x) < 0\).
3. Solve \(f(x) \geq 0\).

8 Completing the square to get the vertex ★★★

Write \(y = x^2 + 8x + 3\) in vertex form and give the vertex.

9 Vertex form with a negative leading coefficient ★★★

Write \(y = -2x^2 + 12x - 7\) in vertex form. What is the maximum value?

10 Factoring by grouping ★★★

Solve \(2x^2 + 5x - 12 = 0\) by factoring.

11 Quadratic formula with a radical ★★★

Solve \(x^2 - 4x - 3 = 0\). Give the exact solutions, then round to the nearest hundredth.

12 A double root ★★★

For which values of \(k\) does \(x^2 + kx + 25 = 0\) have exactly one real solution? Find that solution for each value.

13 Building a function from its roots ★★★

A parabola crosses the x-axis at \(x = 2\) and \(x = 7\) and passes through \((0, 28)\). Find its equation in standard form.

14 Two quadratic inequalities ★★★

Solve each inequality and write the answer in interval notation.

1. \(x^2 - 5x + 4 > 0\)
2. \(2x^2 + x - 6 \leq 0\)

15 The patio ★★★

A rectangular patio is \(4\) feet longer than it is wide. Its area is \(252\) square feet. Find its dimensions.

16 Complex solutions ★★★

Solve \(x^2 - 6x + 13 = 0\) over the complex numbers, then check one of the solutions.

17 A rock thrown from a cliff ★★★

A rock is thrown upward from the top of a \(40\)-foot cliff. Its height in feet after \(t\) seconds is \(h(t) = -16t^2 + 24t + 40\).

1. When does the rock hit the ground?
2. What is its maximum height, and when is it reached?

18 Revenue and price ★★★

A school club sells \(120 - 4p\) T-shirts per week when the price is \(p\) dollars. The weekly revenue is \(R(p) = p(120 - 4p)\).

1. Which price gives the maximum revenue, and what is that revenue?
2. For which prices is the revenue at least \(\$800\)?

19 A line and a parabola ★★★

Find the points where the parabola \(y = x^2 - 2x - 3\) meets the line \(y = x + 1\).

20 Roots that differ by 6 ★★★

The equation \(x^2 - 10x + c = 0\) has two real solutions that differ by \(6\). Find \(c\) and the solutions.

21 Always positive ★★★

For which values of \(k\) is \(x^2 - 4x + k > 0\) true for every real number \(x\)? Explain with the vertex and with the discriminant.