Parabola / Quadratic Equation Question

[SwordfishDragon]

Hey all, need help with this question badly, don't even know where to start. Thanks much.

Determine the quadratic equation in standard form (y=ax²+bx+c) given:

Zeroes at -3 and 7 and an optimal value of 4; using your equation, find the y value, if x = -5

 

[Dr.Peterson]

Hey all, need help with this question badly, don't even know where to start. Thanks much.

Determine the quadratic equation in standard form (y=ax²+bx+c) given:

Zeroes at -3 and 7 and an optimal value of 4; using your equation, find the y value, if x = -5

What have you been taught about quadratic (or perhaps polynomial in general) equations? Do you know the Factor Theorem? What do the two zeros tell you about factors of the function?

Where will the optimal value occur?

Please show us anything at all that you know, even if it is not enough to think that you have started solving the problem. I would give different hints depending on what you have learned.

 

[SwordfishDragon]

What have you been taught about quadratic (or perhaps polynomial in general) equations? Do you know the Factor Theorem? What do the two zeros tell you about factors of the function?

Where will the optimal value occur?

Please show us anything at all that you know, even if it is not enough to think that you have started solving the problem. I would give different hints depending on what you have learned.

I am in Year 10, and in a academic math course. We've currently learned how to factor, expanding, parabolas, zeroes, optimal values, etc.
The problem with this question is that I am not sure how to come up with an equation with the information given.

 

[Otis]

Hi SD. This is the factored form you're working towards:

y = (a)(x - m)(x - n)

where m and n are constants and a is the polynomial's leading coefficient. You can expand it later (after you find values for a,m,n) to get the standard form requested.

As an example, I'm thinking of a different parabola than yours. For my quadratic polynomial, the number -11 is a zero. You already understand polynomial zeros, so you know when x is -11 that my polynomial (called y) will equal zero.

The form above shows that y is a product of three factors. The only way products become zero is if one or more of their factors are zero.

The leading coefficient (a) can never be zero, in a quadratic polynomial. Therefore, at least one of the two binomial factors (x-m) or (x-n) must become zero when x equals -11. What does that tell you about the value of m (or of n)?

 

[SwordfishDragon]

Hi SD. This is the factored form you're working towards:

y = (a)(x - m)(x - n)

where m and n are constants and a is the polynomial's leading coefficient. You can expand it later (after you find values for a,m,n) to get the standard form requested.

As an example, I'm thinking of a different parabola than yours. For my quadratic polynomial, the number -11 is a zero. You already understand polynomial zeros, so you know when x is -11 that my polynomial (called y) will equal zero.

The form above shows that y is a product of three factors. The only way products become zero is if one or more of their factors are zero.

The leading coefficient (a) can never be zero, in a quadratic polynomial. Therefore, at least one of the two binomial factors (x-m) or (x-n) must become zero when x equals -11. What does that tell you about the value of m (or of n)?

I understand but can you help me start this out? Where do I begin? And what would the optimum value determine besides telling me it opens down/up

 

[Dr.Peterson]

I understand but can you help me start this out? Where do I begin? And what would the optimum value determine besides telling me it opens down/up

If you had been given an equation like y = 2(x-4)(x+2), how would you find its zeros and its optimum value? (There are a couple very different ways to do the latter.)

You are just doing the reverse in this problem. If you haven't seen an example like it, you are probably expected to think in reverse. How are the factors related to the zeros? That's the place to begin!

 

[Deleted member 4993]

Hey all, need help with this question badly, don't even know where to start. Thanks much.

Determine the quadratic equation in standard form (y=ax²+bx+c) given:

Zeroes at -3 and 7 and an optimal value of 4; using your equation, find the y value, if x = -5

I would first try to approximately sketch the function.

I'll use the fact that the graph is symmetric about the vertical line at the optimal value.

At what 'x' do we get the optimal value here?

 

[Steven G]

If x=a is a zero, then x-a must be a factor. This is the key point that you need.

 

[Otis]

I understand but can you help me

Hi. If you understood my question, then why did you ignore it?



PS: Here's another question for which we'd like a response: Do you have a textbook that covers this material?

 

[Otis]

If x=a is a zero, then x-a must be a factor.

Note To Readers: Jomo did not intend to use the polynomial's leading coefficient as an example root. (Symbol a is already being used to represent something else, in this exercise.) Let's rephrase the quote above:

If x=m is a polynomial zero, then x-m is a factor of the polynomial.

Additionally, if we also know that x=n is another root of the same polynomial, then we know that x-n is also a factor of the polynomial.

In this exercise, we're told that the polynomial is quadratic, and the two roots are given. Hence, we may immediately write the following equation, after substituting the given root values for m and n.

y = (a)(x - m)(x - n)

The next step is to find the leading coefficient's value.

 

[Steven G]

Note To Readers: Jomo did not intend to use the polynomial's leading coefficient as an example root. (Symbol a is already being used to represent something else, in this exercise.) Let's rephrase the quote above:


Additionally, if we also know that x=n is another root of the same polynomial, then we know that x-n is also a factor of the polynomial.

In this exercise, we're told that the polynomial is quadratic, and the two roots are given. Hence, we may immediately write the following equation, after substituting the given root values for m and n.

y = (a)(x - m)(x - n)

The next step is to find the leading coefficient's value.

Thanks for covering for me!