Answer:
y = (1/10)(x +4)(x +2)(x -3)²
Step-by-step explanation:
We observe that the function intercepts the x-axis at -4, -2, and +3. It crosses the x-axis at the first two of these intercepts, and touches the x-axis at x=3. x-axis intercepts correspond to factors (x -p), where "p" is the x-value of the intercept point.
A factor with odd degree will cause the function to change its sign as the x-value crosses the "p" value. If the degree is even, the sign of the function on either side of "p" is the same.
This means the factors (x+4) and (x+2) will have odd degree, and the factor (x-3) will have even degree. Since there are no "flat" spots* at the intercept points, we take the degrees to be the minimum possible values, 1, 1, and 2.
For some multiplier "a", the function can be written ...
y = a(x +4)(x +2)(x -3)²
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Then for x=0, the value of the function is ...
y = a(4)(2)(-3)² = 72a
We note that the graph appears to have a y-intercept of slightly more than 7, so an appropriate value of "a" is 1/10. That gives a y-intercept of 7.2, a good match to the graph.
Then the factored function is ...
y = (1/10)(x +4)(x +2)(x -3)²
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* When the degree of a factor is more than 1, the derivative of the function at the x-intercept will be zero. This means the graph will have a horizontal tangent ("flat spot") at the x-intercept. The higher the degree of the factor, the "flatter" the graph will be at the x-intercept.
