How do you find the vertex and intercepts for  y = 4x^2 – 6x – 3?

Feb 29, 2016

$\textcolor{b l u e}{\text{ "x_("intercepts")~= 1.896" or " -0.396" to 3 decimal places}}$
color(blue)(" "y_("intercept")=-3)
color(magenta)(" Vertex "->" "(x,y)" "->" "(3/4,-21/4)

Explanation:

I am going to use a method that is the beginning of building the Vertex Equation Form but not taking it all the way.

First observation is that $4 {x}^{2}$ is positive. This means the graph is of shape type $\cup$

(If it had been $- 4 {x}^{2}$ then the shape type would be $\cap$)
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Given:$\text{ } y = 4 {x}^{2} - 6 x - 3$...........................(1)

color(blue)("To determine "x_("vertex"))

Write as:$\text{ } y = 4 \left({x}^{2} - \frac{6}{4} x\right) - 3$

$\textcolor{b r o w n}{\text{I have taken the 4 from "4x" outside the brackets.}}$
$\textcolor{b r o w n}{\text{Note that } 4 \times \left(- \frac{6}{4}\right) x = - 6 x}$

Now consider the $- \frac{6}{4} \text{ from } - \frac{6}{4} x$

Apply this operation: $\left(- \frac{1}{2}\right) \times \left(- \frac{6}{4}\right) = + \frac{3}{4}$

" "color(blue)(x_("vertex") = +3/4

$\textcolor{b r o w n}{\text{With some equations you can do this in your head making the}}$$\textcolor{b r o w n}{\text{process very fast!}}$
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color(blue)("To determine "y_("vertex"))

Substitute$\text{ } \textcolor{g r e e n}{x = \frac{3}{4}}$ into equation (1)

$\textcolor{b r o w n}{y = 4 {x}^{2} - 6 x - 3 \text{ "->" } y = 4 {\left(\textcolor{g r e e n}{\frac{3}{4}}\right)}^{2} - 6 \left(\textcolor{g r e e n}{\frac{3}{4}}\right) - 3}$

color(blue)(" "y_("vertex")=9/4-9/2-3 =- 5 1/4)
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color(blue)("To determine "y_("intercept"))
$\textcolor{b r o w n}{\text{The y intercept is when } x = 0}$

So equation (1) becomes
$y = 4 {\left(0\right)}^{2} - 6 \left(x\right) - 3$
$\textcolor{b l u e}{{y}_{\text{intercept}} = - 3}$

color(magenta)("Vertex "->" "(x,y)" "->" "(3/4,-21/4)

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color(blue)("To determine "x_("intercepts"))

The values for x are non integer values so use the formular
Standard for $y = a {x}^{2} + b x + c$

where $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$a = + 4$
$b = - 6$
$c = - 3$
Giving:

$\text{ } x = \frac{+ 6 \pm \sqrt{{\left(- 6\right)}^{2} - 4 \left(4\right) \left(- 3\right)}}{2 \left(4\right)}$

$\text{ } x = \frac{+ 6 \pm \sqrt{36 + 48}}{8}$

$\text{ } x = \frac{6 \pm \sqrt{{2}^{2} \times 21}}{8}$

$\text{ } x = \frac{6 \pm 2 \sqrt{21}}{8}$

$\textcolor{b l u e}{\text{ "x_("intercepts")~= 1.896" or " -0.396" to 3 decimal places}}$
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