Quadratic Graphs and Features
Understanding parabolas and their properties
What is a Quadratic Function?
A quadratic function has the form \(y = ax^2 + bx + c\), where \(a \neq 0\).
The graph of a quadratic function is called a parabola — a U-shaped curve that is symmetric.
Key Features of a Parabola
Vertex
The highest or lowest point on the parabola
Axis of Symmetry
Vertical line through the vertex: \(x = -\frac{b}{2a}\)
Y-Intercept
Where the graph crosses the y-axis (when x = 0): the point (0, c)
X-Intercepts (Roots)
Where the graph crosses the x-axis (when y = 0)
Effect of the Coefficient 'a'
When a > 0
Parabola opens UPWARD
Vertex is a MINIMUM
When a < 0
Parabola opens DOWNWARD
Vertex is a MAXIMUM
The Larger |a|, the Narrower the Parabola
\(y = 3x^2\) is narrower than \(y = x^2\)
\(y = 0.5x^2\) is wider than \(y = x^2\)
Finding Key Features
Formulas for Key Features
Axis of Symmetry:
\[x = -\frac{b}{2a}\]
Vertex: \(\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\)
Y-intercept: \((0, c)\)
Example 1: Finding All Features
For the quadratic \(y = x^2 - 4x + 3\), find the key features.
\(a = 1, b = -4, c = 3\)
Since \(a = 1 > 0\), parabola opens upward (minimum)
\[x = -\frac{-4}{2(1)} = \frac{4}{2} = 2\]
\[y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1\]
Vertex: \((2, -1)\)
\[(x - 1)(x - 3) = 0\]
\(x = 1\) or \(x = 3\)
X-intercepts: \((1, 0)\) and \((3, 0)\)
Interactive Quadratic Graph Explorer
Explore \(y = ax^2 + bx + c\)
Creating a Table of Values
Example 2: Table for \(y = x^2 - 2x - 3\)
| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| y | 5 | 0 | -3 | -4 | -3 | 0 | 5 |
Notice the symmetry around x = 1 (the axis of symmetry)!
Vertex is at (1, -4) — the minimum point.
The Discriminant and X-Intercepts
Discriminant: \(\Delta = b^2 - 4ac\)
If \(\Delta > 0\): Two distinct x-intercepts
If \(\Delta = 0\): One x-intercept (touches the axis)
If \(\Delta < 0\): No x-intercepts (doesn't cross the axis)
Practice Problems
Question 1: For \(y = x^2 + 6x + 5\), find the vertex and axis of symmetry.
Show Solution
Axis of symmetry: \(x = -\frac{6}{2(1)} = -3\)
Vertex y-value: \((-3)^2 + 6(-3) + 5 = 9 - 18 + 5 = -4\)
Vertex: (-3, -4)
Axis of Symmetry: x = -3
Question 2: Does the parabola \(y = -2x^2 + 4x + 1\) open upward or downward? Is the vertex a maximum or minimum?
Show Solution
Since \(a = -2 < 0\), the parabola opens downward.
The vertex is a maximum.
Question 3: Find the x-intercepts of \(y = x^2 - 5x + 6\).
Show Solution
Solve \(x^2 - 5x + 6 = 0\)
\((x - 2)(x - 3) = 0\)
\(x = 2\) or \(x = 3\)
X-intercepts: (2, 0) and (3, 0)
Question 4: Find the maximum value of \(y = -x^2 + 4x + 5\).
Show Solution
Axis of symmetry: \(x = -\frac{4}{2(-1)} = 2\)
Maximum value: \(y = -(2)^2 + 4(2) + 5 = -4 + 8 + 5 = 9\)
Maximum value is 9 (at x = 2)
Question 5: Use the discriminant to determine how many x-intercepts \(y = x^2 + 2x + 5\) has.
Show Solution
\(\Delta = b^2 - 4ac = (2)^2 - 4(1)(5) = 4 - 20 = -16\)
Since \(\Delta < 0\), the parabola has no x-intercepts.
Quick Reference for CSEC
\(a > 0\): Opens up (minimum) | \(a < 0\): Opens down (maximum)
Axis of Symmetry: \(x = -\frac{b}{2a}\)
Y-intercept: \((0, c)\)
Discriminant \(\Delta = b^2 - 4ac\) tells number of roots