Find the rate of change of f on [x,x+h] if the function is f(x)=3x^2+4x-6

1 answer:

7 0

$\bf slope = m = \cfrac{rise}{run} \implies \cfrac{ f(x_2) - f(x_1)}{ x_2 - x_1}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array}\\\\[-0.35em] \rule{34em}{0.25pt}\\\\ f(x)= 3x^2+4x-6\qquad \begin{cases} x_1=x\\ x_2=x+h \end{cases}\implies \cfrac{f(x+h)-f(x)}{(x+h)-x} \\\\\\ \cfrac{[3(x+h)^2+4(x+h)-6]~~-~~[3x^2+4x-6]}{h} \\\\\\ \cfrac{[3(x^2+2hx+h^2)+4x+4h-6]~~~~-3x^2-4x+6}{h}$

$\bf \cfrac{[\underline{3x^2}+6hx+3h^2\underline{+4x}+4h\underline{-6}]~~~~\underline{-3x^2-4x+6}}{h}\implies \cfrac{6hx+3h^2+4h}{h} \\\\\\ \cfrac{\underline{h}(6x+3h+4)}{\underline{h}}\implies 6x+3h+4$

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The true statement is:

"The range of the function is all real numbers less than or equal to 9."

<h3>Which statements are true?</h3>

Here we have the quadratic function:

$f(x) = -x^2 - 4x + 5$

And we want to see which of the given statements are true.

The first one is:

"The domain is al real numbers less than or equal to -2"

This is false, for all quadratic functions the domain is the set of all real numbers (unless the domain is defined).

The second statement is:

" The domain of the function is all real numbers less than or equal to 9."

Also false

Third one:

" The range of the function is all real numbers less than or equal to −2"

The range of a quadratic function with a negative leading coefficient will be the set of all the values smaller than the y-value of the vertex.

In this case, the quadratic function is:

$f(x) = -x^2 - 4x + 5$

So the vertex is at:

$x = 4/(2*-1) = -2\\$