Section 3.3. Analyzing Graphs of Quadratic Functions
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1 Section 3.3 Analyzing Graphs of Quadratic Functions
2 Introduction
3 Definitions A quadratic function is a function with the form f (x) = ax 2 + bx + c, where a 0.
4 Definitions A quadratic function is a function with the form f (x) = ax 2 + bx + c, where a 0. The graphs of quadratic functions are all parabolas - informally, they have a bowl or a U shape, either upside down or right-side up. a > 0 We say the parabola opens up. a < 0 We say the parabola opens down.
5 Vertex Form of a Quadratic
6 Definitions Every quadratic can be written in vertex form : f (x) = a(x h) 2 + k
7 Definitions Every quadratic can be written in vertex form : f (x) = a(x h) 2 + k In this form, (h, k) is the vertex of the parabola (and a still determines if the parabola opens up or down): (h, k)
8 Definitions Every quadratic can be written in vertex form : f (x) = a(x h) 2 + k In this form, (h, k) is the vertex of the parabola (and a still determines if the parabola opens up or down): axis of symmetry x = h (h, k) The parabola is always symmetric across the line x = h, which is the vertical line that goes through the vertex.
9 Putting an Equation into Vertex Form The vertex of the parabola f (x) = ax 2 + bx + c is given by: h = b 2a k = f (h) 1. Calculate h and k using the formulas, and get a from the original equation.
10 Putting an Equation into Vertex Form The vertex of the parabola f (x) = ax 2 + bx + c is given by: h = b 2a k = f (h) 1. Calculate h and k using the formulas, and get a from the original equation. 2. Plug these into f (x) = a(x h) 2 + k
11 Examples Write the following quadratic functions into vertex form: 1. f (x) = x 2 4x + 5
12 Examples Write the following quadratic functions into vertex form: 1. f (x) = x 2 4x + 5 f (x) = (x 2) 2 + 1
13 Examples Write the following quadratic functions into vertex form: 1. f (x) = x 2 4x + 5 f (x) = (x 2) f (x) = 3x 2 + 4x + 1
14 Examples Write the following quadratic functions into vertex form: 1. f (x) = x 2 4x + 5 f (x) = (x 2) f (x) = 3x 2 + 4x + 1 f (x) = 3 ( x 2 3)
15 Maximums and Minimums of Quadratics
16 Absolute Maximums and Minimums An absolute maximum or maximum is a point on the graph that is higher than every other point.
17 Absolute Maximums and Minimums An absolute maximum or maximum is a point on the graph that is higher than every other point. An absolute minimum or minimum is a point on the graph that is lower than every other point.
18 Absolute Maximums and Minimums An absolute maximum or maximum is a point on the graph that is higher than every other point. An absolute minimum or minimum is a point on the graph that is lower than every other point. This has an absolute minimum but no absolute maximum. This has an absolute maximum but no absolute minimum.
19 Maximums and Minimums on a Quadratic Every quadratic has either a maximum or a minimum at its vertex. (h, k) (h, k) If a > 0, it has a minimum. If a < 0, it has a maximum.
20 Examples 1. Mendoza Manufacturing plans to produce a one-compartment vertical file by bending the long side of a 10-in by 18-in. sheet of plastic along two lines to form a shape. How tall should the file be in order to maximize the volume it can hold, and what is the maximum volume?
21 Examples 1. Mendoza Manufacturing plans to produce a one-compartment vertical file by bending the long side of a 10-in by 18-in. sheet of plastic along two lines to form a shape. How tall should the file be in order to maximize the volume it can hold, and what is the maximum volume? A height of 4.5 in gives the files its maximum volume of 405 in 3.
22 Examples (continued) 2. A soft-drink vendor determines its revenue and costs are determined by the function R(x) = 10x C(x) = 0.002x x where x is the number of drinks sold. Find the vendor s maximum profit.
23 Examples (continued) 2. A soft-drink vendor determines its revenue and costs are determined by the function R(x) = 10x C(x) = 0.002x x where x is the number of drinks sold. Find the vendor s maximum profit. $7100
24 Graphing Quadratics
25 Method 1. Find the vertex of the parabola.
26 Method 1. Find the vertex of the parabola. 2. Find the x and y intercepts.
27 Method 1. Find the vertex of the parabola. 2. Find the x and y intercepts. 3. Plot all of the above points and connect them with a parabola.
28 Example For the function f (x) = 2x 2 7x 4, find the vertex, the x and y-intercepts, the axis of symmetry, the maximum or minimum, and then graph.
29 Example For the function f (x) = 2x 2 7x 4, find the vertex, the x and y-intercepts, the axis of symmetry, the maximum or minimum, and then graph. Vertex: ( 7 4, 81 ) 8 x-intercepts: ( 1 2, 0), (4, 0) y-intercept: (0, 4) Axis of symmetry: x = 7 4 Minimum:
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