Answer:
- f(x) = 2x(x -4)
- f(x) = 2(x -2)² +(-8)
Explanation:
The factors of the factored form of the equation for the quadratic can be found by reading the x-intercept from the graph. The vertex form can be found by reading the coordinates of the vertex. The leading coefficient can be found any of several ways.
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factored form
The graph crosses the x-axis at x=0 and x=4. For x-intercept 'p', (x -p) is a factor. The leading coefficient can be found by matching the function value to a point on the graph somewhere other than at an x- or y-intercept.
If we choose to match the value at x=2, we must have f(2) = -8.
f(2) = a(x)(x -4) = a(2)(2 -4) = -8
-4a = -8 . . . . simplify
a = 2 . . . . . divide by -4
The factored form is ...
f(x) = 2(x)(x -4)
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vertex form
We can read the vertex from the graph as (2, -8). We note that 1 unit either side of the vertex, the graph rises 2 units, to (1, -6) and (3, -6). This tells us the leading coefficient is a=+2.
The vertex form equation is ...
f(x) = a(x -h)² +k
where the vertex is (h, k) and 'a' is the leading coefficient. For the values we read from the graph, the vertex-form equation is ...
f(x) = 2(x -2)² +(-8)