Question

Can you help me create a polynomial function matching the criteria on the left please and thanks

Answer 1

Answer:

$f(x) = \dfrac{1}{8}(x + 4)^2(x + 1)(x - 3)$

Step-by-step explanation:

From inspection of the graph, we can see that the curve intercepts the x-axis at (-4, 0), (-1, 0) and (3, 0)

Therefore,

x = -4 ⇒ x + 4 = 0

x = -1 ⇒ x + 1 = 0

x = 3 ⇒ x - 3 = 0

Because (-4, 0) touches the x-axis, then (x + 4)² will be a factor

So (x + 4)², (x + 1) and (x - 3) are all factors of the polynomial

$\implies f(x) = \dfrac{1}{n}(x + 4)^2(x + 1)(x - 3)$

If we multiply the constants, this will give us the y-intercept:

⇒ 4² x 1 x -3 = -48

From inspection of the graph, the y-intercept is -6

So to get from -48 to -6 we need to multiply -48 by 1/8

Therefore, n = 1/8

$\implies f(x) = \dfrac{1}{8}(x + 4)^2(x + 1)(x - 3)$

Answer 2

Step-by-step explanation:

the factored representation has at least one major advantage : the zero-solutions are directly visible.

as whenever one factor turns zero, the whole function turns zero.

the zero solutions are

(-4, 0)

(-1, 0)

(3, 0)

based on this we get

(x + 4)(x + 1)(x - 3)

for x = -4 the first factor is 0.

for x = -1 the second factor is 0.

for x = 3 the third factor is zero.

the y-intercept gives us a hint about 1/#, as this is the y value when x = 0.

this is then

1/# × (0+4)(0+1)(0-3) = 1/# × -12

the point (0, -6) tells us this must be equal to -6.

1/# × -12 = -6

1/# = -6/-12 = 1/2

# = 2

so, we get

f(x) = 1/2 × (x + 4)(x + 1)(x - 3)

but that is still not the whole result.

why ?

because the first zero solution (-4, 0) is actually the merger of 2 zero solutions, as the vertex of a loop is only touching the x-axis.

if we shift the whole graph a little bit down, we see that this x-intercept would turn into 2 intercepts.

so, this function has actually 4 zero solutions.

that tells us that the polynomial is of 4th degree (there must be a "x⁴" somewhere in the expression as highest exponent of x).

but multiplying only 3 factors based on x will give us only x³ as highest x exponent.

the solution : the first factor has to represent 2 zero solutions as the graph indicates.

how ?

by squaring that factor.

so, our function looks like

f(x) = 1/2 × (x + 4)²(x + 1)(x - 3)

now we get "x⁴" in the expression.

but wait, now for x = 0 the 1/# is not right anymore.

we have now

1/# × (0+4)²(0+1)(0-3) = 1/# × 16×1×-3 = 1/# × -48

so,

1/# × -48 = -6

1/# = -6/-48 = 1/8

# = 8

so, finally, the fully correct function is

f(x) = 1/8 × (x + 4)²(x + 1)(x - 3)