Question

What are the important points needed to graph `f(x) = -(x-2)(x + 5)`?

Answer 1

This is an instruction/guide to the method needed, No direct values for your equation are given.

Explanation:

This is a quadratic and there are a few tricks that may be used to find salient points for sketching them.

Given: `y=-(x-2)(x+5)`

Multiply the brackets giving:

`y = -x^2-3x+10`....... (1)

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First off; we have a negative `x^2`. This results in an inverted horse shoe type plot. That is of shape `nn` instead of U.

Using standard form of `y=ax^2+bx+c`
To do the next bit you would need to change this standard form into `y=a(x^2 +b/a x + c/a)`. It is the bit inside the brackets we are looking at. In your case `a=1` so we do not need to change anything.
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`color(blue)("The minima for "x "occurs at " -1/2 times b/a")`
`color(blue)("In your case")`
`color(blue)(a=1)`
`color(blue)(b=-3)`

so `color(red)(x_("minimum") = (-1/2) times (-3) = + 3/2)`

Substitute `color(red)(x_("minimum"))` in equation (1) giving

`color(red)(y= -(3/2)^2-3(3/2)+10 )`

`color(green)("You have now found the values for " (x,y)_("minimum"))`
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`color(blue)(" To find y-intercept substitute "x=0" in equation (1)")`
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`color(blue)(" To find x-intercepts substitute "y=0" in equation (1)")`
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