Question

Please help I am terrible at algebra

Answer 1

Step-by-step explanation:

PART 1.

A quadratic function is given by the form:

`f(x)=ax^2+bx+c`

While a linear function is given by:

`y=mx+b`

In this case, we have 2 cases:

CASE 1

`f(x)=-x(2x+8)`

Applying distributive property we get:

`f(x)=-x(2x+8) \\ \\ f(x)=-2x^2-8x`

So this is a quadratic function.

`\text{Quadratic term:}-2x^2 \\ \\ \text{linear term:}-8x \\ \\ \text{constant term:} None`

CASE 2

`f(x)=x^2-7`

So this is a quadratic function.

`\text{Quadratic term:} \ x^2 \\ \\ \text{linear term:} \ None \\ \\ \text{constant term:} \ -7`

PART 2.

Remember that for a quadratic function we have:

`f(x)=ax^2+bx+c \\ \\ \\ a,b,c \ are \ constants`

From the table:

`When \ x=0, \ y=9: \\ \\ f(0)=9=a(0)^2+b(0)+c \\ \\ c=9`

`When \ x=1, \ y=16: \\ \\ f(1)=16=a(1)^2+b(1)+9 \\ \\ 16=a+b+9 \\ \\ a+b=7`

`When \ x=-4, \ y=1: \\ \\ f(-4)=1=a(-4)^2+b(-4)+9 \\ \\ 1=16a-4b+9 \\ \\ 16a-4b=-8`

Solving for a and b:

`a=7-b \\ \\ \text{Substituting into 16a-4b=-8:} \\ \\ 16(7-b)-4b=-8 \\ \\ 112-16b-4b=-8 \\ \\ -20b=-8-112 \\ \\-20b=-120 \\ \\ b=6 \\ \\ Then: \\ \\ a=7-6 \\ \\ a=1`

So the equation is:

`\boxed{f(x)=x^2+6x+9}`