Question

Find a rule for a quadratic function that opens up and has x-intercepts of (4,0) and (-3,0) and contains point (5,16)

Answer 1

have you covered quadratics yet? do you know how to find a solution to a quadratic equation?

Answer 2

I have, but i don't fully understand. I missed school today, and we have the final tomorrow. We went over this in class today, so i missed it. I'd say i know how to find it, but its all in my notes, which are at school.

Answer 3

"has x-intercepts of (4,0) and (-3,0)"
meaning it has 2 solutions, that is x = 4 and x = -3
recall, solutions are when we set y=0 and solve for "x", that is when the graph touches the x-axis or the x-intercepts

at those x-intercepts, x = 4 and x = -3 thus

so we can say that \(\bf y={\color{red}{ a}}(\square x^2+\square x+\square )\\
x = 4\implies x-4=0\implies {\color{blue}{ (x-4)}}=0
\\ \quad \\
x = -3\implies x+3=0\implies {\color{blue}{ (x+3)}}=0
\\ \quad \\
{\color{red}{ a}}{\color{blue}{ (x-4)}}{\color{blue}{ (x+3)}}=0
\\ \quad \\
{\color{red}{ a}}{\color{blue}{ (x-4)}}{\color{blue}{ (x+3)}}=\textit{original polynomial}\)

Answer 4

so if we multiply those 2 binomials, we end up with \(\bf y={\color{red}{ a}}(\square x^2+\square x+\square )
\\ \quad \\ \quad \\
{\color{red}{ a}}(x^2-x-12)\)

so... what's the multiplier "a"? well
we also know that the parabola is going upwards, that is, it opens up
that means that "a" is positive for one

and we also know that the parabola also passes through the point (5, 16)
meaning that when x=5 y = 16
so let us plug that in the equation thus \(\bf y={\color{red}{ a}}(\square x^2+\square x+\square )
\\ \quad \\ \quad \\

y={\color{red}{ a}}(x^2-x-12)\qquad x=5\qquad y=16\implies 16={\color{red}{ a}}(5^2-5-12)\)

solve for "a" see what you get
then plug it in the equation

Answer 5

So here's what i got...
16=a(5^2−5−12)
=a(25-5-12)
=a(20-12)
=a(8)
16=8(5^2-5-12)
Or would it just be : 16=a(8) ?

Answer 6

yeap, that means \(\bf y={\color{red}{ a}}(x^2-x-12)\qquad x=5\qquad y=16\implies 16={\color{red}{ a}}(5^2-5-12)
\\ \quad \\
16=8a\implies 2=a\qquad thus\qquad y={\color{red}{ 2}}(x^2-x-12)\)

that'd be equation

Answer 7

Thanks so much! :)

Answer 8

you can distribute if you want it, or leave it like so

Answer 9

yw