Week8Notes

Quadratic Functions (Vertex Form)

• Recall: Different graph transformations:

  • Vertical stretches

  • Vertical shifts

  • Horizontal shifts

Definition of Quadratic Functions

• A quadratic function is a transformed version of the basic function f(x) = x².

• The vertex is the point where the graph changes direction, starting at (0,0).

• Example: g(x) = 2x²

  • Represents a vertical stretch by a factor of 2.

  • Note: The graph is symmetric about the y-axis since (-x)² = x².

Graph Transformations

• Example: g(x) = x²

  • Inverting the function results in vertical flips (downward curve).

  • Example: g(x) = -0.5x²

    • Indicates a vertical flip and compression of the graph.

• General form: g(x) = ax²

  • Where 'a' controls the curve direction (a > 0 ⇒ upward; a < 0 ⇒ downward).

Horizontal Shifts and Vertex Form

• Example: g(x) = 2(x-3)² + 4

  • (x-3) indicates a rightward shift by 3 units.

  • +4 indicates an upward shift by 4 units.

  • Vertex moves to (3, 4).

• General form in vertex form: g(x) = a(x - h)² + k

  • Vertex at (h, k).

Identifying Vertex and Optimization

• Critical Concept: The vertex indicates either a maximum or minimum.

  • If a < 0, the graph curves down (maximum).

  • If a > 0, the graph curves up (minimum).

• Example (Profit Maximization):

  • Profit function P(x) = -3(x-40)² + 500.

  • Maximum profit occurs at vertex (40, 500).

  • Hence, maximum profit is $500 at x = 40.

Cost Minimization Example

• Example: Cost function C(x) = 7(x-10)² + 20

  • Lowest cost occurs at vertex (10, 20).

  • Thus, minimum cost is $20 at x = 10.

Solving Quadratic Equations

• To solve for f(x) = some number, first set f(x) = 0.

• Example: Solving f(x) = -x² + 9:

  • f(x) = 0 → -x² + 9 = 0 → x² = 9 → x = ±3.

  • Indicates two possible solutions: x = 3 or x = -3.

Additional Examples

• For f(x) = (x+7)² = -19:

  • Solve (x+7)² = 19 → x + 7 = ±√19 → x = -7 ± √19.

• Symmetry around the vertex observed where h is between the two zeroes.

Generalizing Solutions

• To find when f(x) equals a number other than zero, rearrange:

  • Example: Solve -2x² + 17 = 3 by subtracting.

  • Example: Solve 3(x+2)² + 1 = 9 by simplifying.

Converting Between Forms

• Standard Form Definition: Quadratics in standard form f(x) = ax² + bx + c.

• Conversions:

  • From vertex form: f(x) = a(x-h)² + k leads to standard form via expansion.

  • From standard to vertex form involves completing the square.

Practical Applications: Maximizing Revenue

• Example (Running a Business):

  • Demand function D(p)=300-7p, Revenue R(p) = p(300-7p).

  • Resulting standard form: R(p) = -7p² + 300p.

  • Calculate vertex to find max revenue = $3214.86 at price p = 21.429.

Profit Function Analysis

• Profit(P) = Revenue - Cost.

• Example of determining max profit through function transformations.

• Use quadratic formula for direct calculations of zeroes: Zeroes = -b ± √(b²-4ac) / 2a.

Factored Form

• Quadratics can be expressed as f(x) = a(x-x1)(x-x2).

• Zeroes are easily identified, e.g., f(x) = 2(x-4)(x+3): Zeroes at x = 4, x = -3.

• If no real zeroes exist, such as in g(t) = 2(t-5)² + 10, the factored form does not materialize.