# 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