# How do you find the vertex and intercepts for y=x^2+2x-1?

Jan 19, 2016

Vertex$\to \left(x , y\right) \to \left(- 1 , - 2\right)$
x-axis intercepts $\to x \approx 0.414 \text{ or } x \approx - 2.414$ to 3 decimal places
y-axis intercept $= - 1$

#### Explanation:

Given: $y = {x}^{2} + 2 x - 1$ ....................................(1)

$\textcolor{b l u e}{\text{To determine the vertex.}}$

This is already in the form of $y = a \left({x}^{2} + \frac{b}{a} x - 1\right)$ because a=1

${x}_{\text{vertex}} = \left(- \frac{1}{2}\right) \times \frac{b}{a}$
This is a variation on 'Vertex Form Equation'
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$\textcolor{b l u e}{{x}_{\text{vertex}} = \left(- \frac{1}{2}\right) \times \left(+ 2\right) = - 1}$......................(2)

Substitute (2) into equation (1)

${y}_{\text{vertex}} = {\left(- 1\right)}^{2} + 2 \left(- 1\right) - 1$

$\textcolor{b l u e}{{y}_{\text{vertex}} = + 1 - 2 - 1 = - 2}$

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$\textcolor{b l u e}{\text{To determine "y_("intercept}}$

The graph crosses the y-axis when $x = 0$

So $y = {x}^{2} + 2 x - 1 \to {y}_{\text{intercept}} = {\left(0\right)}^{2} + 2 \left(0\right) - 1$

color(blue)(y_("intercept")=-1
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$\textcolor{b l u e}{\text{To determine "x_("intercepts}}$

The graph crosses the x-axis when $y = 0$

So $y = 0 = {x}^{2} + 2 x - 1$

We need the form y=0=(x+?)(x+?) to find any integer factors

The only factor of 1 is 1 and to get -1 we need $\left(- 1\right) \times \left(+ 1\right)$
which is fine until you try and get $+ 2 x$. It fails then! So it means that the factors are not integer values. In which case we need to use the formula.
Using: $\textcolor{w h i t e}{\ldots \ldots} y = a {x}^{2} + b x + c$

Where $\textcolor{w h i t e}{\ldots} x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

a=1 ; b=+2 ; c=-1

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

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

but $\sqrt{8} = \sqrt{2 \times {2}^{2}} = 2 \sqrt{2}$

$x = \frac{- 2 \pm 2 \sqrt{2}}{2}$

$x \approx 0.414 \text{ or } x \approx - 2.414$ to 3 decimal places