How do you find the vertex and the intercepts for f(x) = -x^2 + 2x + 5?

Jun 6, 2017

it vertex is maximum at $\left(1 , 6\right)$

x-intercepts $= 1 \pm \sqrt{6}$

y-intercept $= 5$

Explanation:

$f \left(x\right) = - {x}^{2} + 2 x + 5$

$f \left(x\right) = - \left({x}^{2} - 2 x\right) + 5$

$f \left(x\right) = - {\left(x - 1\right)}^{2} + {\left(- 1\right)}^{2} + 5$

$f \left(x\right) = - {\left(x - 1\right)}^{2} + 6$

since coefficient of $\left(x - 1\right)$ is -ve value, it vertex is maximum at $\left(1 , 6\right)$

to find x-intercept, plug in $f \left(x\right) = 0$ in the equation, therefore
$0 = - {\left(x - 1\right)}^{2} + 6$
${\left(x - 1\right)}^{2} = 6$
$x - 1 = \pm \sqrt{6}$
$x = 1 \pm \sqrt{6}$

to find y-intercept, plug in $x = 0$ in the equation, therefore
$f \left(0\right) = - {\left(0 - 1\right)}^{2} + 6$
$f \left(0\right) = - {\left(- 1\right)}^{2} + 6$
$f \left(0\right) = - 1 + 6 = 5$

Jun 6, 2017

$\text{see explanation}$

Explanation:

$\text{for the standard form of a parabola } y = a {x}^{2} + b x + c$

"then " x_(color(red)"vertex")=-b/(2a)

$y = - {x}^{2} + 2 x + 5 \text{ is in this form}$

$\text{with " a=-1,b=2" and } c = 5$

$\Rightarrow {x}_{\textcolor{red}{\text{vertex}}} = - \frac{2}{- 2} = 1$

$\text{substitute this value into function for y-coordinate}$

$\Rightarrow {y}_{\textcolor{red}{\text{vertex}}} = - {\left(1\right)}^{2} + \left(2 \times 1\right) + 5 = 6$

$\Rightarrow \textcolor{m a \ge n t a}{\text{vertex }} = \left(1 , 6\right)$

$\textcolor{b l u e}{\text{ for intercepts}}$

• " let x = 0, in function for y-intercept"

• " let y = 0, in function for x-intercepts"

$x = 0 \to y = 0 + 0 + 5 = 5 \leftarrow \textcolor{red}{\text{ y-intercept}}$

$y = 0 \to - {x}^{2} + 2 x + 5 = 0$

$\text{solve using the "color(blue)"quadratic formula}$

$x = \frac{- 2 \pm \sqrt{4 + 20}}{- 2} = \frac{- 2 \pm \sqrt{24}}{-} 2 = \frac{- 2 \pm 2 \sqrt{6}}{-} 2$

$\Rightarrow x = 1 \pm \sqrt{6}$

$\Rightarrow x \approx 3.45 , x \approx - 1.45 \leftarrow \textcolor{red}{\text{ x-intercepts}}$
graph{-x^2+2x+5 [-12.65, 12.66, -6.34, 6.3]}