Graphing Quadratic Functions

What is a Quadratic Function?

A quadratic function has the form:

f(x) = ax² + bx + c or y = ax² + bx + c

where a ≠ 0.

The graph of a quadratic function is a parabola - a U-shaped curve.

Key features:

• Opens up (if a > 0) or down (if a < 0)
• Has a vertex (highest or lowest point)
• Has an axis of symmetry (vertical line through vertex)
• May have 0, 1, or 2 x-intercepts

The Standard Form

Standard Form: y = ax² + bx + c

What each coefficient tells us:

a (leading coefficient):

• If a > 0: parabola opens UP (∪)
• If a < 0: parabola opens DOWN (∩)
• |a| > 1: narrower parabola (steeper)
• |a| < 1: wider parabola (flatter)

c (constant term):

• The y-intercept (where graph crosses y-axis)
• The point (0, c)

b (middle coefficient):

• Affects the position of the vertex horizontally
• Used to find axis of symmetry

Parent Function: y = x²

The simplest quadratic is y = x².

Key points:

• Vertex at origin (0, 0)
• Opens upward
• Axis of symmetry: x = 0 (y-axis)
• Symmetric about y-axis

Table of values: | x | y = x² | |----|--------| | -2 | 4 | | -1 | 1 | | 0 | 0 | | 1 | 1 | | 2 | 4 |

The parabola is symmetric: same y-values for ±x

The Vertex

The vertex is the turning point of the parabola.

If a > 0: Vertex is the MINIMUM (lowest point)

If a < 0: Vertex is the MAXIMUM (highest point)

Finding the vertex from y = ax² + bx + c:

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

y-coordinate: Substitute x-value into equation

Example: Find vertex of y = x² - 4x + 3

x = -(-4)/(2(1)) = 4/2 = 2

y = (2)² - 4(2) + 3 = 4 - 8 + 3 = -1

Vertex: (2, -1)

Axis of Symmetry

The axis of symmetry is a vertical line through the vertex.

Equation: x = -b/(2a)

This is the same as the x-coordinate of the vertex!

Example: y = 2x² + 8x + 5

Axis of symmetry: x = -8/(2·2) = -8/4 = -2

The parabola is symmetric about the line x = -2.

Finding the Y-Intercept

The y-intercept is where the graph crosses the y-axis (when x = 0).

Simply evaluate f(0) = c

Example: y = 2x² - 5x + 3 Y-intercept: (0, 3)

Finding X-Intercepts (Zeros/Roots)

The x-intercepts are where the graph crosses the x-axis (when y = 0).

Set y = 0 and solve: ax² + bx + c = 0

Use factoring, quadratic formula, or other methods.

Example: Find x-intercepts of y = x² - 5x + 6

Set y = 0: x² - 5x + 6 = 0 Factor: (x - 2)(x - 3) = 0 x = 2 or x = 3

X-intercepts: (2, 0) and (3, 0)

Number of X-Intercepts

Use the discriminant b² - 4ac:

b² - 4ac > 0: Two x-intercepts (parabola crosses x-axis twice)

b² - 4ac = 0: One x-intercept (parabola touches x-axis at vertex)

b² - 4ac < 0: No x-intercepts (parabola doesn't reach x-axis)

Example 1: y = x² - 4 Discriminant: 0² - 4(1)(-4) = 16 > 0 Two x-intercepts: x = ±2

Example 2: y = x² + 1 Discriminant: 0² - 4(1)(1) = -4 < 0 No x-intercepts

Graphing Using Key Points

Step-by-step process:

Step 1: Determine if parabola opens up or down (check sign of a)

Step 2: Find vertex using x = -b/(2a), then find y

Step 3: Draw axis of symmetry (vertical line through vertex)

Step 4: Find y-intercept (0, c)

Step 5: Find x-intercepts if they exist (solve ax² + bx + c = 0)

Step 6: Plot additional points if needed

Step 7: Draw smooth U-shaped curve through points

Example 1: Complete Graphing

Graph: y = x² - 4x + 3

Step 1: a = 1 > 0, opens UP

Step 2: Find vertex x = -(-4)/(2·1) = 2 y = (2)² - 4(2) + 3 = -1 Vertex: (2, -1)

Step 3: Axis of symmetry: x = 2

Step 4: Y-intercept: (0, 3)

Step 5: X-intercepts: x² - 4x + 3 = 0 (x - 1)(x - 3) = 0 x = 1 or x = 3 Points: (1, 0) and (3, 0)

Step 6: Use symmetry If (0, 3) is on the graph, then (4, 3) is also on it (symmetric about x = 2)

Step 7: Draw parabola through: (1,0), (2,-1), (3,0), (0,3), (4,3)

Example 2: Parabola Opening Down

Graph: y = -x² + 2x + 3

Step 1: a = -1 < 0, opens DOWN

Step 2: Vertex x = -2/(2·(-1)) = 1 y = -(1)² + 2(1) + 3 = 4 Vertex: (1, 4) - this is the maximum!

Step 3: Axis of symmetry: x = 1

Step 4: Y-intercept: (0, 3)

Step 5: X-intercepts: -x² + 2x + 3 = 0 Multiply by -1: x² - 2x - 3 = 0 (x - 3)(x + 1) = 0 Points: (-1, 0) and (3, 0)

Step 6: Plot and draw downward parabola

Vertex Form

Vertex form: y = a(x - h)² + k

where (h, k) is the vertex.

This form makes graphing easier!

Converting from vertex form to standard form: Expand the squared binomial

Example: y = 2(x - 3)² + 1

Vertex: (3, 1) Opens up (a = 2 > 0) Narrower than y = x² (a = 2 > 1)

Expand to standard form: y = 2(x² - 6x + 9) + 1 y = 2x² - 12x + 18 + 1 y = 2x² - 12x + 19

Transformations from y = x²

y = x² + k: Vertical shift

• k > 0: shift UP k units
• k < 0: shift DOWN |k| units

y = (x - h)²: Horizontal shift

• h > 0: shift RIGHT h units
• h < 0: shift LEFT |h| units

y = ax²: Vertical stretch/compression

• a > 1: vertical stretch (narrower)
• 0 < a < 1: vertical compression (wider)
• a < 0: reflection over x-axis

Example: y = (x - 2)² + 3

Starting from y = x²:

• Shift right 2 units
• Shift up 3 units
• Vertex moves from (0,0) to (2,3)

Maximum and Minimum Values

If a > 0 (opens up):

• Minimum value at vertex
• No maximum value

If a < 0 (opens down):

• Maximum value at vertex
• No minimum value

Example: y = -2x² + 8x - 3

Vertex: x = -8/(2·(-2)) = 2 y = -2(2)² + 8(2) - 3 = -8 + 16 - 3 = 5

Since a < 0, maximum value is 5 (at x = 2)

Domain and Range

Domain: All quadratic functions have domain: all real numbers (-∞, ∞) or {x | x ∈ ℝ}

Range: Depends on vertex and direction

If a > 0 (opens up): Range: [k, ∞) where k is y-coordinate of vertex

If a < 0 (opens down): Range: (-∞, k] where k is y-coordinate of vertex

Example: y = x² - 4x + 3 Vertex: (2, -1), opens up Range: [-1, ∞)

Comparing Parabolas

Width: Determined by |a|

• Larger |a|: narrower parabola
• Smaller |a|: wider parabola

Example:

• y = 3x² is narrower than y = x²
• y = (1/2)x² is wider than y = x²
• y = -x² and y = x² have same width (|a| = 1)

Using a Table of Values

When in doubt, make a table!

Example: y = x² + 2x - 3

| x | y | Point | |----|------|----------| | -3 | 0 | (-3, 0) | | -2 | -3 | (-2, -3) | | -1 | -4 | (-1, -4) | | 0 | -3 | (0, -3) | | 1 | 0 | (1, 0) |

Vertex appears to be at (-1, -4) Verify: x = -2/(2·1) = -1 ✓

Applications: Projectile Motion

Height of projectile: h(t) = -16t² + v₀t + h₀

The graph is a parabola opening down.

Example: Ball thrown at 48 ft/s from ground level h(t) = -16t² + 48t

Maximum height at vertex: t = -48/(2·(-16)) = 1.5 seconds h = -16(1.5)² + 48(1.5) = 36 feet

The ball reaches 36 feet at 1.5 seconds.

Applications: Business

Profit/Revenue functions often quadratic.

Example: Profit = -2x² + 40x - 50 where x is number of items (in hundreds)

Maximum profit at vertex: x = -40/(2·(-2)) = 10 P = -2(10)² + 40(10) - 50 = 150

Maximum profit is $15,000 (when x = 10 hundreds = 1000 items)

Applications: Area

Example: You have 100 feet of fence. What dimensions maximize the rectangular area?

Let x = width Then (100 - 2x)/2 = length = 50 - x

Area: A = x(50 - x) = -x² + 50x

Maximum at vertex: x = -50/(2·(-1)) = 25 feet Length = 25 feet

Maximum area: 625 square feet (a square!)

Sketching vs. Precise Graphing

Quick sketch needs:

• Vertex
• Direction (up/down)
• Y-intercept
• General shape

Precise graph needs:

• Vertex
• Axis of symmetry
• Y-intercept
• X-intercepts
• Additional points
• Scale marked clearly

Common Mistakes to Avoid

1. Wrong direction Check sign of a carefully!

2. Forgetting negative in vertex formula x = -b/(2a), not b/(2a)

3. Not simplifying vertex coordinates Show actual numbers, not formulas

4. Assuming parabola crosses x-axis Check discriminant!

5. Plotting points but not drawing smooth curve Use a smooth U-shape, not line segments

6. Forgetting axis of symmetry Use it to find additional points easily

Quick Reference

Standard Form: y = ax² + bx + c

Vertex: (-b/(2a), f(-b/(2a)))

Axis of Symmetry: x = -b/(2a)

Y-intercept: (0, c)

X-intercepts: Solve ax² + bx + c = 0

Direction:

• a > 0: opens up
• a < 0: opens down

Width:

• |a| > 1: narrow
• |a| < 1: wide

Practice Strategy

Level 1: Simple parabolas

• y = x²
• y = x² + 3
• y = -x²

Level 2: Shifted parabolas

• y = (x - 2)²
• y = (x + 1)² - 4

Level 3: Standard form with factoring

• y = x² - 4x + 3

Level 4: Need quadratic formula

• y = x² + 4x + 1

Level 5: Applications

• Projectile motion
• Business optimization
• Area problems

Tips for Success

• Always find the vertex first
• Use axis of symmetry to find symmetric points
• Check if parabola opens up or down
• Calculate discriminant to know about x-intercepts
• Plot at least 5 points for accuracy
• Draw smooth curves, not choppy lines
• Label all key points clearly
• Include arrows at ends of parabola
• Practice vertex formula until automatic