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Math 20 Pre-Calculus P20.7 Demonstrate understanding of quadratic functions of the form y=ax²+bx+c and of their graphs, including: vertex, domain and range, direction of, opening, axis of symmetry, x- and y-intercepts. C. Quadratic Functions

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Key Terms:

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Quadratic Functions occur in a wide variety of real world situations. In this unit we will investigate functions and use them in math modelling and problem solving.

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1. Vertex Form P20.7 Demonstrate understanding of quadratic functions of the form y=ax²+bx+c and of their graphs, including: vertex domain and range direction of opening axis of symmetry x- and y-intercepts.

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1. Vertex Form Investigate p. 143

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The graph of a Quadratic Function is a parabola When the graph opens up the vertex is the lowest point and when it opens down the vertex is the highest point

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The y-coordinate of the vertex is called the min value or max value depending of which way it opens. The parabola is symmetrical about a line called the axis of symmetry. The line divides the graph into two equal halves, left and right. So if you know the a of s and a point you can find another point (unless the point is the vertex)

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The A of S intersects the vertex The x-coordinate of the vertex is the equation of the A of S.

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Quadratic Function in vertex form f(x) = a(x-p) 2 +q are very easy to graph. a, p, and q tell you what you need. (p,q) = Vertex Opens up +a Opens down –a Larger a = narrower parabola Smaller a = wider parabola

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Example 1

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Example 2

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Example 3

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Example 4

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Key Ideas p.156

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Practice Ex. 3.1 (p.157) #1-14 #4-18

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2. Standard Form P20.7 Demonstrate understanding of quadratic functions of the form y=ax²+bx+c and of their graphs, including: vertex domain and range direction of opening axis of symmetry x- and y-intercepts.

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2. Standard Form

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Recall that the Standard form of a quadratic function is f(x) = ax 2 +bx+c or y = ax 2 +bx+c Where a, b, c are real numbers and a ≠ 0 a determines width of graph (smaller a = wider graph) and opening (+a up and –a down) b shifts the graphs left and right c shifts the graph up and down

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We can expand f(x) = a(x-p) 2 +q to get f(x) = ax 2 +bx+c, which will allow us to see the relation between the variable coefficients in each.

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So, b = -2ap or And c = ap 2 + q or q = c – ap 2

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Recall that to determine the x-coordinate of the vertex, you use x = p. So the x-coordinate of the vertex is

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Example 1

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Example 2

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Example 3

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Key Ideas p.173

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Practice Ex. 3.2 (p.174) #1-9, 11-17 odds #5-25 odds

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3. Completing the Square P20.7 Demonstrate understanding of quadratic functions of the form y=ax²+bx+c and of their graphs, including: vertex domain and range direction of opening axis of symmetry x- and y-intercepts.

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3. Completing the Square You can express a quadratic function in vertex form, f(x) = a(x-p) 2 +q or standard form f(x) = ax 2 +bx+c We already know we can go from vertex to standard by just expanding However to graph by hand it is much easier if the function is in vertex form because we have the vertex, axis of symmetry and max or min of the graph

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So to be able to turn a standard form function into vertex form would be advantageous. This process is called Completing the Square

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What we want to be able to do is rewrite the trinomial as a binomial squared. (x+5)(x+5) = (x+5) 2

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Lets complete the square:

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If there is a coefficient in front of the x 2 term we have to add a couple steps. Complete the Square:

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Example 1

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Example 2

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Example 3

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Example 4

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Key Ideas p.192

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Practice Ex. 3.3 (p.192) #1-9, 10-18 evens #1-9 odds in each, 10-28 evens

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