Training Quadratic and Polynomial Functions Graphing Quadratics — Vertex Form
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Graphing Quadratics — Vertex Form

22 min Quadratic and Polynomial Functions

Vertex Form of a Parabola

Every quadratic function y = ax² + bx + c graphs as a parabola — a smooth U-shaped curve that opens upward when a is positive and downward when a is negative. The vertex of the parabola is its highest or lowest point, making it essential for optimization problems.

The vertex form y = a(x − h)² + k makes the vertex immediately visible: it sits at the point (h, k). Converting between standard and vertex form, often by completing the square, is a key skill for both graphing and problem solving.

This lesson covers graphing parabolas, finding the vertex, identifying the axis of symmetry, and determining maximum or minimum values.

Vertex Form

$$f(x) = a(x - h)^2 + k$$

Vertex: $(h, k)$. Axis of symmetry: $x = h$.

  • $a > 0$: opens up (vertex is minimum)
  • $a < 0$: opens down (vertex is maximum)
  • $|a| > 1$: narrower than $y = x^2$
  • $|a| < 1$: wider than $y = x^2$
Standard → Vertex

Given $f(x) = ax^2 + bx + c$:

$$h = -\frac{b}{2a}, \quad k = f(h)$$

Example 1

Find the vertex of $f(x) = 2x^2 - 12x + 22$.

$h = -\dfrac{-12}{2(2)} = 3$. $k = 2(9) - 36 + 22 = 4$.

Vertex: $(3, 4)$. Vertex form: $f(x) = 2(x - 3)^2 + 4$.

Example 2

Identify vertex, direction, axis, and intercepts of $f(x) = -(x + 1)^2 + 9$.

Vertex: $(-1, 9)$. Opens down ($a = -1$). Axis: $x = -1$.

$y$-int: $f(0) = -1 + 9 = 8$.

$x$-int: $0 = -(x+1)^2 + 9 \Rightarrow (x+1)^2 = 9 \Rightarrow x = 2$ or $x = -4$.

Example 3

$f(x) = x^2 + 4x + 7$. Find vertex, direction, and range.

$h = -2$. $k = 4 - 8 + 7 = 3$. Vertex: $(-2, 3)$. Opens up.

Range: $[3, \infty)$. No $x$-intercepts ($\Delta = 16 - 28 < 0$).

Graphing Checklist

Steps
  1. Find vertex $(h, k)$.
  2. Draw axis of symmetry $x = h$.
  3. Find $y$-intercept: $f(0)$.
  4. Find $x$-intercepts (if real).
  5. Plot symmetric points and sketch.

Practice Problems

1. Vertex of $f(x) = x^2 - 6x + 5$
2. Vertex of $f(x) = -2(x + 3)^2 + 7$
3. Direction of $f(x) = -x^2 + 4x$
4. $y$-intercept of $f(x) = 3(x - 1)^2 - 4$
5. $x$-intercepts of $f(x) = x^2 - 4x - 5$
6. Axis of symmetry of $f(x) = 2x^2 + 8x + 3$
7. Convert $f(x) = x^2 - 10x + 21$ to vertex form
8. Range of $f(x) = -(x - 2)^2 + 5$
9. Max or min value of $f(x) = 3x^2 - 12x + 7$
10. Does $f(x) = x^2 + 2x + 5$ have real $x$-intercepts?
Show Answer Key

1. $(3, -4)$

2. $(-3, 7)$

3. Down ($a = -1 < 0$)

4. $f(0) = 3(1) - 4 = -1$

5. $x = 5$ and $x = -1$

6. $x = -2$

7. $(x - 5)^2 - 4$

8. $(-\infty, 5]$

9. Min value at $x = 2$: $f(2) = 12 - 24 + 7 = -5$

10. $\Delta = 4 - 20 = -16 < 0$. No.

The Quadratic Formula and Discriminant Polynomial Functions — Degree and End Behavior