Graphing Quadratic Equations
A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the form:
Where a, b, and c are constants, and a ≠ 0. When graphed, a quadratic equation forms a parabola.
Key Features of a Parabola
Standard Form and Vertex Form
Standard form: f(x) = ax² + bx + c
Vertex form: f(x) = a(x - h)² + k
Where (h, k) represents the vertex of the parabola.
| Feature | Description | How to Find |
|---|---|---|
| Vertex | The highest or lowest point of the parabola | x = -b/(2a), then find y by substituting x |
| Axis of Symmetry | Vertical line passing through the vertex | x = -b/(2a) |
| y-intercept | Point where the parabola crosses the y-axis | f(0) = c |
| x-intercepts | Points where the parabola crosses the x-axis (roots) | Use the quadratic formula: x = (-b ± √(b² - 4ac))/(2a) |
| Direction | Opens upward or downward | If a > 0, opens upward; if a < 0, opens downward |
- If discriminant > 0: Two real roots (x-intercepts)
- If discriminant = 0: One real root (parabola touches x-axis at one point)
- If discriminant < 0: No real roots (parabola doesn't cross x-axis)
Methods for Graphing Quadratic Equations
Method 1: Point Plotting
- Choose several x-values.
- Calculate corresponding y-values using the equation.
- Plot the points and connect them with a smooth curve.
Example: Graph f(x) = x² - 2x - 3
Pick values: x = -2, -1, 0, 1, 2, 3, 4
Calculate y-values:
- f(-2) = (-2)² - 2(-2) - 3 = 4 + 4 - 3 = 5
- f(-1) = (-1)² - 2(-1) - 3 = 1 + 2 - 3 = 0
- f(0) = (0)² - 2(0) - 3 = 0 - 0 - 3 = -3
- f(1) = (1)² - 2(1) - 3 = 1 - 2 - 3 = -4
- f(2) = (2)² - 2(2) - 3 = 4 - 4 - 3 = -3
- f(3) = (3)² - 2(3) - 3 = 9 - 6 - 3 = 0
- f(4) = (4)² - 2(4) - 3 = 16 - 8 - 3 = 5
Plot these points and connect with a smooth curve to see the parabola.
Method 2: Finding Key Features
- Find the vertex using x = -b/(2a) and substituting to find y.
- Determine the axis of symmetry (same x-value as the vertex).
- Find the y-intercept by evaluating f(0).
- Find x-intercepts using the quadratic formula.
- Plot these key points and draw the parabola.
Example: Graph f(x) = x² - 6x + 8
Step 1: Find vertex
x = -b/(2a) = -(-6)/(2*1) = 6/2 = 3
y = f(3) = 3² - 6(3) + 8 = 9 - 18 + 8 = -1
Vertex: (3, -1)
Step 2: Axis of symmetry
x = 3
Step 3: y-intercept
f(0) = 0² - 6(0) + 8 = 8
y-intercept: (0, 8)
Step 4: x-intercepts
Using the quadratic formula:
x = (-(-6) ± √((-6)² - 4(1)(8)))/(2(1))
x = (6 ± √(36 - 32))/2
x = (6 ± √4)/2
x = (6 ± 2)/2
x = 4 or x = 2
x-intercepts: (2, 0) and (4, 0)
Method 3: Converting to Vertex Form
Converting from standard form f(x) = ax² + bx + c to vertex form f(x) = a(x - h)² + k
- Factor out 'a' from the first two terms: f(x) = a(x² + (b/a)x) + c
- Complete the square for the expression inside parentheses.
- Rewrite in vertex form.
Example: Convert f(x) = 2x² - 12x + 16 to vertex form
Step 1: Factor out 'a'
f(x) = 2(x² - 6x) + 16
Step 2: Complete the square
To complete the square for (x² - 6x), we add and subtract (-6/2)² = 9
f(x) = 2(x² - 6x + 9 - 9) + 16
f(x) = 2((x - 3)² - 9) + 16
f(x) = 2(x - 3)² - 18 + 16
Step 3: Rewrite in vertex form
f(x) = 2(x - 3)² - 2
This confirms the vertex is at (3, -2)
Effects of Constants a, b, and c
The Effect of 'a'
The coefficient 'a' affects the width and direction of the parabola:
- If |a| > 1: The parabola is narrower (steeper)
- If 0 < |a| < 1: The parabola is wider (flatter)
- If a > 0: The parabola opens upward (U-shape)
- If a < 0: The parabola opens downward (∩-shape)
The Effect of 'b'
The coefficient 'b' affects the axis of symmetry and the location of the vertex:
- As |b| increases, the vertex moves further from the y-axis
- The sign of 'b' affects the direction the vertex moves
The Effect of 'c'
The constant 'c' is the y-intercept of the parabola:
- It shifts the entire parabola up or down
- Increasing 'c' shifts the parabola upward
- Decreasing 'c' shifts the parabola downward
Interactive Quadratic Function Grapher
Adjust the coefficients to see how they affect the graph:
Equation: f(x) = x² + 0x + 0
Special Cases and Applications
Quadratic with No x-intercepts
Example: f(x) = x² + 4
This parabola never crosses the x-axis because the discriminant (b² - 4ac) = 0² - 4(1)(4) = -16 < 0.
Quadratic with One x-intercept (Tangent to x-axis)
Example: f(x) = x² - 6x + 9 = (x - 3)²
This parabola touches the x-axis at exactly one point (x = 3) because the discriminant is 0.
Applications to Motion
A ball thrown upward has a height-time relationship described by:
h(t) = -16t² + v₀t + h₀
Where v₀ is initial velocity and h₀ is initial height.