# How do you find the important points to graph y=-x^2-3x+2?

Aug 14, 2018

See explanation

#### Explanation:

Given: $y = - {x}^{2} - 3 x + 2$
Compare to $y = a {x}^{2} + b x + c$

As the ${x}^{2}$ term in negative the general shape is $\cap$

$\textcolor{red}{\text{The y-intercept "=c=+2 ->" point} \left(x , y\right) \to \left(0 , 2\right)}$

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Write in the form $y = a \left({x}^{2} + \frac{b}{a} x\right) + c$

In this case $a = - 1 \mathmr{and} b = - 3$ giving:

$y = - 1 \left({x}^{2} + 3 x\right) + 2$

x_("vertex")=[color(white)(".")(-1/2)xx(b/a)color(white)(".")] -> [color(white)(".")(-1/2)xx(-3)/(-1)color(white)(".")] = -3/2

So by substitution:

${y}_{\text{vertex}} = - {\left(- \frac{3}{2}\right)}^{2} - 3 \left(- \frac{3}{2}\right) + 2$

${y}_{\text{vertex}} = - \frac{9}{4} + \frac{9}{2} + 2 = 4 \frac{1}{4} \to \frac{17}{4}$

$\textcolor{red}{\text{Vertex} \to \left(x , y\right) = \left(- \frac{3}{2} , \frac{17}{4}\right)}$
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Notice that $\left(- 1\right) \times \left(- 2\right) = + 2 \leftarrow c$
and that $\left(- 1\right) + \left(- 2\right) = - 3 \leftarrow b$

so initially we would think that we have the factorisation. However the negative ${x}^{2}$ gives us a problem. Perhaps we can 'force' our initial thoughts to give us the correct form.

Set y=(x-1)(x-2)color(white)("dd") ->color(white)("dd") y=x^2-3x+2 larr" Fail"

Lets try:
y=(-x-1)(x-2)color(white)("dd")->color(white)("dd") y=-x^2+x+2larr" Fail"

Ok! Looks as though we do not have whole number factorisation. So lets use the formula $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$y = \textcolor{w h i t e}{\text{dd.d}} a {x}^{2} + b x + c$
$y = \left(- 1\right) {x}^{2} - 3 x + 2$

So a=-1; b=-3 and c=+2 giving:

$x = \frac{+ 3 \pm \sqrt{{\left(- 3\right)}^{2} - 4 \left(- 1\right) \left(+ 2\right)}}{2 \left(- 1\right)}$

$x = - \frac{3}{2} \pm \frac{\sqrt{17}}{2} \leftarrow \text{ Exact solution}$

As 17 is a prime number we can not simplify this any further.

Approximate solution:

$x = - 1.5 \pm 2.062$ to 3 decimal places giving:

color(red)(x_("intercept")~~ -3.562 and +0.562" to 3 decimal places")