Algebra 2 Forms of Quadratic Functions Lesson (Free PDF)

Q: What is vertex form of a quadratic?

Vertex form is f(x) = a(x - h)² + k. The vertex is (h, k), the axis of symmetry is x = h, and k is the maximum when a is negative or the minimum when a is positive.

Q: How do you complete the square?

Take half of the x-coefficient, square it, and add and subtract it. If a is not 1, factor a out first. The result is a perfect square plus a constant.

Q: How do you find the vertex from standard form?

Use x = -b/(2a) for the x-coordinate and substitute into the function for the y-coordinate. Or complete the square to get vertex form.

Q: Which form shows the zeros?

Factored form a(x - r)(x - s) shows the zeros r and s directly, because each factor equal to zero gives a zero.

Q: How can you tell if a parabola has a maximum or minimum?

Look at the sign of a. If a is positive the parabola opens up and has a minimum at the vertex. If a is negative it opens down and has a maximum.

Algebra 2 math lesson: Forms of Quadratic Functions – Maths PDF

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This quadratic forms lesson for Algebra 2 covers forms of quadratic functions (Common Core A-SSE.B.3a, A-SSE.B.3b, F-IF.C.8a), with a free PDF to print.

A quadratic function can be written in standard form, factored form, or vertex form, and each form reveals a different feature. Standard form shows the y-intercept, factored form shows the zeros, and vertex form a(x – h)² + k shows the vertex. Completing the square converts standard form to vertex form.

Algebra 2 · Common Core State Standards: A-SSE.B.3a, A-SSE.B.3b, F-IF.C.8a

Key facts in this quadratic forms lesson

Key facts

Standard form ax² + bx + c: y-intercept c.
Factored form a(x – r)(x – s): zeros r and s.
Vertex form a(x – h)² + k: vertex (h, k) and extreme value k.
Complete the square: add and subtract (b/2a)².
a > 0 opens up (minimum); a < 0 opens down (maximum).

What are the three forms of a quadratic function?

A quadratic function can be written in three equivalent forms. Each form makes a different feature of the parabola easy to read.

Form	Equation	What it shows
Standard	\(f(x) = ax^{2} + bx + c\)	y-intercept \(c\); axis \(x = -\frac{b}{2a}\)
Factored	\(f(x) = a(x – r)(x – s)\)	zeros \(r\) and \(s\)
Vertex	\(f(x) = a(x – h)^{2} + k\)	vertex \((h, k)\); max or min \(k\)

f(x) = (x – 2)² – 4 = x(x – 4): vertex (2, -4), zeros 0 and 4, axis x = 2.

The sign of \(a\) gives the direction: positive opens up (the vertex is a minimum) and negative opens down (a maximum). The size of \(a\) controls the width: a larger \(|a|\) makes a narrower parabola, and a smaller \(|a|\) makes a wider one.

Positive a opens up (minimum); negative a opens down (maximum).

How do you complete the square?

Completing the square rewrites standard form as vertex form. The area model shows the idea: \(x^{2} + 6x\) is a square of side \(x\) plus two strips of area \(3x\). Adding the corner \(9\) completes a square of side \(x + 3\).

x² + 6x + 9 = (x + 3)²: the missing corner is 9.

If \(a \ne 1\), factor \(a\) out of the \(x^{2}\) and \(x\) terms.
Take half of the \(x\)-coefficient and square it.
Add and subtract that square, and account for any factor outside.
Write the perfect square trinomial as a square and combine constants.

Example: \(x^{2} – 8x + 5\). Half of \(-8\) is \(-4\), and \((-4)^{2} = 16\). So \(x^{2} – 8x + 16 – 16 + 5 = (x – 4)^{2} – 11\). The vertex is \((4, -11)\).

Each conversion reveals a different feature.

The forms are equal: for instance \((x – 2)^{2} – 4 = x^{2} – 4x = x(x – 4)\). Choose the form that answers the question: zeros, extreme value, or intercept.

Quadratic forms lesson: questions and answers

What is vertex form of a quadratic?

Vertex form is f(x) = a(x – h)² + k. The vertex is (h, k), the axis of symmetry is x = h, and k is the maximum when a is negative or the minimum when a is positive.

How do you complete the square?

Take half of the x-coefficient, square it, and add and subtract it. If a is not 1, factor a out first. The result is a perfect square plus a constant.

How do you find the vertex from standard form?

Use x = -b/(2a) for the x-coordinate and substitute into the function for the y-coordinate. Or complete the square to get vertex form.

Which form shows the zeros?

Factored form a(x – r)(x – s) shows the zeros r and s directly, because each factor equal to zero gives a zero.

How can you tell if a parabola has a maximum or minimum?

Look at the sign of a. If a is positive the parabola opens up and has a minimum at the vertex. If a is negative it opens down and has a maximum.

How to use this quadratic forms lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

Download the quadratic forms lesson as a PDF with the button below and keep it with your notes.

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