find the rule of correspondence for a quadratic function whose graph has the indicated features. Write the rule in standard form.

1. x- intercepts at (-3,0) and (1,0); the same shape as y=x^2

2.x- intercepts at (7/2,0) and (5,0); the same shape as y= -3x^2

Update:

what do you mean this is straight out of the textbook the question, I'm trying to figure out how you don't understand what rule of correspondence means.

Relevance

➊ For y = ax² + bx + c

If h and k are zeros of the quadratic function, then (x-h) and (x-k)

are factors of it ... zeros being the value(s) of x that make

the function zero, i.e. the value(s) of x that produce y=0.

Notice that this doesn't allow for all coefficients 'a' of x²:

➋ y = (x-h)(x-k)  =  x² -hx - kx + hk  =  x² + (-h-k)x + hk            ← b = (-h-k)

c = hk

But, 'a' can only be 1.

So, we need to incorporate 'a' into ➋.

That modifies ➋ to the final form

➌ y = a(x-h)(x-k)

Now, using ➌, your problems become easy to do.

——————————————————————————————————————

1. x-intercepts at (-3,0) and (1,0); the same shape as y = x².

Note:

x-intercepts are the zeros of the function ... i.e. 1 & -3 are zeros

and the same shape as y = x² indicate that a=1.

So,

➌ y = a(x-h)(x-k)  =  1(x-1)(x-(-3))  =  (x-1)(x+3)  =  x²+2x-3

——————————————————————————————————————

2. x-intercepts at (7/2,0) and (5,0); the same shape as y= -3x²

Note

x-intercepts are   ⁷∕₂ & 5   and   a =  -3

So,

➌ y = a(x-h)(x-k)  =   -3(x-⁷∕₂)(x-5)  =  (x-⁷∕₂)(x-5)  =  -3x²+(⁵¹∕₂)x - ¹⁰⁵∕₂

y  =  -3x²+(⁵¹∕₂)x - ¹⁰⁵∕₂    ← ANSWER

Hope this is what you are looking for.

Have a good one!

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