Graphing Quadratic Functions

What is a Quadratic Function?

A quadratic function has the form: f(x) = ax² + bx + c

Graph: A parabola (U-shaped curve)

Key features:

• Vertex (highest or lowest point)
• Axis of symmetry (line of symmetry)
• y-intercept
• x-intercepts (if any)

Standard Form

f(x) = ax² + bx + c

The value of a determines:

• If a > 0: Opens upward (U shape)
• If a < 0: Opens downward (∩ shape)
• |a| larger: Narrower parabola
• |a| smaller: Wider parabola

Example: Direction

f(x) = 2x²: Opens upward (a = 2 > 0)

f(x) = -x²: Opens downward (a = -1 < 0)

Parent Function: f(x) = x²

Simplest quadratic function

Key points:

• Vertex: (0, 0)
• Axis of symmetry: x = 0 (y-axis)
• Opens upward
• y-intercept: (0, 0)

Table of values:

Symmetric about y-axis

Vertex Form

f(x) = a(x - h)² + k

Where:

• (h, k) is the vertex
• a determines direction and width
• x = h is axis of symmetry

Easiest form for graphing!

Example 1: Identify Vertex

f(x) = (x - 3)² + 2

Compare to f(x) = a(x - h)² + k:

• h = 3, k = 2

Vertex: (3, 2) Axis of symmetry: x = 3 Opens: Upward (a = 1)

Example 2: Negative Values

f(x) = -(x + 2)² - 1

Rewrite: f(x) = -1(x - (-2))² + (-1)

Vertex: (-2, -1) Axis of symmetry: x = -2 Opens: Downward (a = -1)

Graphing from Vertex Form

Steps:

1. Identify vertex (h, k)
2. Plot vertex
3. Find additional points using symmetry
4. Draw parabola

Example: Graph f(x) = (x - 1)² - 4

Step 1: Vertex = (1, -4)

Step 2: Axis of symmetry: x = 1

Step 3: Find points on one side

Step 4: Mirror across x = 1

Step 5: Draw smooth curve through points

Finding Vertex from Standard Form

For f(x) = ax² + bx + c:

Vertex x-coordinate: h = -b/(2a)

Vertex y-coordinate: k = f(h) (substitute x = h)

Example: Find Vertex

f(x) = x² + 6x + 5

Step 1: Find h

• a = 1, b = 6
• h = -6/(2×1) = -3

Step 2: Find k

• k = f(-3) = (-3)² + 6(-3) + 5
• k = 9 - 18 + 5 = -4

Vertex: (-3, -4)

Axis of symmetry: x = -3

Finding Intercepts

y-intercept: Set x = 0

• f(0) = c
• Point: (0, c)

x-intercepts: Set f(x) = 0

• Solve ax² + bx + c = 0
• Use factoring or quadratic formula

Example: Find Intercepts

f(x) = x² - 4

y-intercept:

• f(0) = 0² - 4 = -4
• Point: (0, -4)

x-intercepts:

• x² - 4 = 0
• (x + 2)(x - 2) = 0
• x = -2 or x = 2
• Points: (-2, 0) and (2, 0)

Transformations

Starting with f(x) = x²:

Vertical shift: f(x) = x² + k

• Up k units if k > 0
• Down k units if k < 0

Horizontal shift: f(x) = (x - h)²

• Right h units if h > 0
• Left h units if h < 0

Vertical stretch/compression: f(x) = ax²

• Narrower if |a| > 1
• Wider if 0 < |a| < 1

Reflection: f(x) = -x²

• Flips over x-axis

Example: Transformations

f(x) = 2(x - 3)² + 1

Compared to x²:

• Right 3 units (h = 3)
• Up 1 unit (k = 1)
• Narrower (a = 2 > 1)

Vertex: (3, 1)

Maximum and Minimum Values

If parabola opens upward (a > 0):

• Vertex is minimum point
• Minimum value = k

If parabola opens downward (a < 0):

• Vertex is maximum point
• Maximum value = k

Example: Find Maximum

f(x) = -2(x + 1)² + 5

Opens: Downward (a = -2) Vertex: (-1, 5)

Maximum value: 5 (at x = -1)

Comparing Two Parabolas

Example: Compare Functions

f(x) = x² and g(x) = (x - 2)² + 3

f(x):

• Vertex: (0, 0)
• Opens upward

g(x):

• Vertex: (2, 3)
• Opens upward
• Same width as f(x)

g(x) is f(x) shifted right 2, up 3

Real-World Applications

Projectile motion: Ball thrown upward

• h(t) = -16t² + 64t + 3
• Vertex gives maximum height and time

Profit function: Business revenue

• P(x) = -2x² + 80x - 200
• Vertex gives maximum profit

Area optimization: Fencing problems

• Find dimensions that maximize area

Architecture: Parabolic arches and bridges

Practice