Quadratic Functions Worksheet with Answers – Algebra 1 (Free PDF)

Problem 1 Read the Vertex Form, Opening Up ★★★

Look at y = 2(x − 3)² + 5. Does its parabola open upward or downward? Name the turning point, the mirror line and the lowest or highest value of y.

Problem 2 Read the Vertex Form, Opening Down ★★★

The rule y = −(x + 4)² − 2 describes a parabola. Decide its direction of opening and name its turning point and its extreme value.

Problem 3 Axis from Standard Form ★★★

Locate the mirror line and the turning point of y = x² − 18x + 50.

Problem 4 Complete a Table of Values ★★★

Build a table of values for y = x² − 6x + 5 using x = 0, 1, 2, 3, 4, 5, 6. Use the table to name the turning point and the point on the y-axis.

Problem 5 Read the Factored Form ★★★

The function y = (x − 2)(x + 6) is in factored form. Read off where its graph meets the x-axis, then find the mirror line and the turning point.

Problem 6 Read a Parabola Graph ★★★

Study the parabola drawn below. Give its highest point, its mirror line, its largest y-value and where it meets the y-axis.

Problem 7 Vertex Form to Standard Form ★★★

Write y = 3(x − 1)² − 4 in standard form.

Problem 8 Vertex of a Downward Parabola ★★★

Find the highest point of the parabola y = −2x² + 12x − 7 and state its greatest y-value.

Problem 9 Complete the Square, Easy Case ★★★

Rewrite y = x² + 10x + 16 as y = a(x − h)² + k and read the turning point. Then find where the graph meets the x-axis.

Problem 10 Completing the Square, No Zeros ★★★

Rewrite y = x² − 8x + 19 as a square plus a constant. Explain why its graph stays above the x-axis.

Problem 11 Completing the Square, a = 2 ★★★

Complete the square for y = 2x² − 12x + 13 and name the turning point.

Problem 12 Graph from Zeros and Vertex ★★★

Sketch y = x² − 2x − 3 on the grid. Before plotting, find where it meets each axis and where it turns.

Problem 13 Vertex from Factored Form ★★★

Where does y = −2(x + 1)(x − 5) turn? Decide whether that point is a maximum or a minimum.

Problem 14 Compare Three Parabolas ★★★

Compare the graphs of y = x², y = 3x² and y = −0.5x². Which is narrowest, which opens down and which is widest?

Problem 15 Ball Launched from a Roof ★★★

From a rooftop, a water balloon is tossed upward. After t seconds its height is h(t) = −16t² + 48t + 64 feet. Find where it starts, how high it climbs and when it lands.

Problem 16 Largest Rectangular Pen ★★★

Dana encloses a rectangular dog run with 96 feet of fence. With width w, the enclosed area is A(w) = w(48 − w). Which width encloses the most ground, and how much is that?

Problem 17 Best Ticket Price ★★★

A drama club sells T-shirts at p dollars each and takes in R(p) = p(250 − 5p) dollars. Which price brings in the most money? How much is that, and at which prices does the club take in nothing?

Problem 18 Equation from Vertex and a Point ★★★

A parabola turns at (−1, 6) and also contains the point (2, −3). Express it as y = a(x − h)² + k and then in standard form.

Problem 19 Equation from Zeros and a Point ★★★

A parabola meets the x-axis at −2 and 4 and contains the point (0, −16). Give its equation in the form y = a(x − p)(x − q).

Problem 20 Compare Two Quadratic Functions ★★★

Let f(x) = x² − 6x + 11. A second function g has the table below.

Which of the two functions dips lower, and at which x does each reach its lowest point?