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

Nov 4, 2015

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: $y = - \left(x - 2\right) \left(x + 5\right)$

Multiply the brackets giving:

$y = - {x}^{2} - 3 x + 10$....... (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 $\cap$ instead of U.

Using standard form of $y = a {x}^{2} + b x + c$
To do the next bit you would need to change this standard form into $y = a \left({x}^{2} + \frac{b}{a} x + \frac{c}{a}\right)$. 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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$\textcolor{b l u e}{\text{The minima for "x "occurs at " -1/2 times b/a}}$
$\textcolor{b l u e}{\text{In your case}}$
$\textcolor{b l u e}{a = 1}$
$\textcolor{b l u e}{b = - 3}$

so $\textcolor{red}{{x}_{\text{minimum}} = \left(- \frac{1}{2}\right) \times \left(- 3\right) = + \frac{3}{2}}$

Substitute $\textcolor{red}{{x}_{\text{minimum}}}$ in equation (1) giving

$\textcolor{red}{y = - {\left(\frac{3}{2}\right)}^{2} - 3 \left(\frac{3}{2}\right) + 10}$

color(green)("You have now found the values for " (x,y)_("minimum"))
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$\textcolor{b l u e}{\text{ To find y-intercept substitute "x=0" in equation (1)}}$
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$\textcolor{b l u e}{\text{ To find x-intercepts substitute "y=0" in equation (1)}}$
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