Algebra 1 Graphing Quadratic Functions and Transformations Lesson (Free PDF)

This quadratic functions lesson for Algebra 1 covers graphing quadratic functions and transformations (Common Core F-IF.C.7a, F-IF.C.8a, F-BF.B.3, A-SSE.B.3b), with a free PDF to print.

A quadratic function f(x) = ax² + bx + c has a U-shaped graph called a parabola. The vertex lies on the axis of symmetry x = -b/2a. In vertex form f(x) = a(x – h)² + k, the vertex is (h, k). Changing k, h, and a shifts, stretches, or reflects the graph.

Algebra 1 · Common Core State Standards: F-IF.C.7a, F-IF.C.8a, F-BF.B.3, A-SSE.B.3b

Key facts in this quadratic functions lesson

What does the graph of a quadratic function look like?

How do you find the vertex and intercepts?

1. The axis of symmetry is \(x = -\frac{b}{2a}\). The vertex has this x-coordinate.
2. Substitute that x-value to find the y-coordinate of the vertex.
3. The y-intercept is \((0, c)\). Find x-intercepts by factoring or the quadratic formula.
4. Plot the vertex, the intercepts, and symmetric points, then draw a smooth curve.

Example: For \(y = x^{2} – 4x + 3\), the axis is \(x = 2\), the vertex is \((2, -1)\), the y-intercept is 3, and the x-intercepts are 1 and 3.

What does vertex form tell you?

In \(f(x) = a(x – h)^{2} + k\), the vertex is \((h, k)\), the axis is \(x = h\), and \(a\) controls the width and direction. Completing the square turns standard form into vertex form: \(x^{2} + 6x + 5\) becomes \((x + 3)^{2} – 4\).

How do transformations change the graph?

Starting from \(f(x) = x^{2}\): adding \(k\) shifts the graph up (positive) or down (negative); replacing \(x\) by \(x – h\) shifts it right when \(h\) is positive; multiplying by \(a\) stretches it, and a negative \(a\) reflects it over the x-axis.

For a quadratic model, the vertex gives the best value. If \(R = -5p^{2} + 100p\) is revenue, the vertex \((10, 500)\) gives the price that earns the most.

Tip: To sketch a parabola quickly, plot the vertex first, then two points on each side at equal distances from the axis of symmetry.

Quadratic functions lesson: questions and answers

How to use this quadratic functions 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 functions lesson as a PDF with the button below and keep it with your notes.

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