 # 3.3 Factored Form of a Quadratic Relation

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3.3 Factored Form of a Quadratic Relation
When you look at an equation, how do you know it’s quadratic? If it’s quadratic, how do you know it’s a parabola? If you have y = x2, of course you know it’s quadratic and a parabola, but what if it’s not so obvious?

Example #1 - Reasoning Is the graph of y = 2(x+1)(x-5) a parabola? If so, in what direction does it open? Explain. x y 1st diff 2nd diff -3 32 - -2 14 14-32=-18 -1 -14 -14-(-18)=4 -10 4 1 -16 -6 2 -18 3 2nd differences are the same so it is quadratic. Since 2nd differences are positive, parabola opens upwards

Example #2 - Strategy Determine the y-intercept, zeros, axis of symmetry, and vertex of the quadratic relation y = 2(x-4)(x+2). Then sketch the graph. 1. y-intercept: x=0 y = 2(x-4)(x+2) y = 2(0-4)(0+2) y = 2(-4)(2) y = -16 Therefore, the y-intercept occurs at (0,-16)

Example #2 – Strategy (cont’d)
2. To find the zeros (x-intercepts), let y=0: 0 = 2(x-4)(x+2) x-4 = 0 or x+2 = 0 x = 4 or x = -2 Therefore, the zeros occur at (4,0) and (-2,0). Recall, axis of symmetry passes through the midpoint of the zeros

Example #3 – Strategy Determine the y-intercept, zeros, axis of symmetry and the vertex of the quadratic relation y = (x-2)2. Then sketch the graph. y-intercept (when x=0) y = (x-2)2 y = (0-2)2 y = (-2)2 y = 4 Therefore, the y-intercept occurs at (0,4).

Example #3 – Strategy cont’d
Zeros (occur where y=0) y = (x-2)2 0 = (x-2)2 Take the square root of both sides: √0 = √[(x-2)2] 0 = (x-2) 0 = x-2 x = 2 Therefore, the zero occurs at (2,0).

Example #3 – Strategy cont’d
Recall: Previously, to find the axis of symmetry, we would look at the midpoint of the two zeros. Since there’s only one zero, the axis of symmetry must pass through it Thus, the axis of symmetry occurs at x=2. Since there is only one zero, this must also be the minimum/vertex, so the vertex occurs at (0,2).

Example #3 – Strategy cont’d
y = (x-2)2

Example #4 Determine an equation for this parabola. Locate the zeros.
(-2,0) and (1,0) We know that a quadratic equation will have the form: y=a(x - r)(x - s) r and s are the zeros Substitute the zeros in: y=a(x – (-2))(x - 1) y=a(x +2)(x - 1)

Example #4 cont’d y=a(x +2)(x - 1)
We find ‘a’ by using the y-intercept: (0,10) 10=a(0 +2)(0 - 1) 10=a(2)(-1) 10=a(-2) Divide both sides by (-2) to isolate ‘a’ 10 −2 = 𝑎×(−2) −2 a = -5

Example #4 cont’d y = -5(x+2)(x-1) Check that it makes sense:
The vertical stretch factor is -5: the negative implies that it opens down, and the 5 has an absolute value greater than 1, so it is vertically stretched The zeros occur at x = -2 and x = +1 Everything checks out.

In Summary… When a quadratic relation is expressed in factored form y = a(x – r)(x – s), each factor can be used to determine a zero, or x- intercept of the parabola An equation for a parabola can be determined using the zeros and the coordinates of one other point on the parabola