How do you graph y=2x^2 -5x -3?

Sep 20, 2015

You can easily do this by inserting values in x and plotting some points.

Explanation:

You can graph this easily by making a table of x and y values. But first, we must find the vertex $\left(h , k\right)$.

To solve for the vertex, we'll start with this formula ($h$ is the value of the abscissa of the vertex.):
$h = - \frac{b}{2 a}$
$h = - \frac{\left(- 5\right)}{2 \left(2\right)}$
$h = \frac{5}{4}$

Now to find $k$ (the ordinate of the vertex), we will simply plug in $\frac{5}{4}$ to $x$.

$y = 2 {x}^{2} - 5 x - 3$
$y = 2 {\left(\frac{5}{4}\right)}^{2} - 5 \left(\frac{5}{4}\right) - 3$
$y = 2 \left(\frac{25}{16}\right) - \frac{25}{4} - 3$
$y = \frac{25}{8} - \frac{25}{4} - 3$
$y = \frac{25}{8} - \frac{50}{8} - 3$
$y = - \frac{25}{8} - 3$
$y = - \frac{25}{8} - \frac{24}{8}$
$y = - \frac{49}{8}$

The vertex is $\left(\frac{5}{4} , - \frac{49}{8}\right)$.

After plotting the vertex, just plug in other values into x and graph them. Start with simple ones such as the y-intercept (set $x$ to $0$). Remember that this is a quadratic equation, so your graph must be a parabola.

graph{2x^2-5x-3 [-14.24, 14.24, -7.12, 7.12]}

Sep 20, 2015

y= (2x+1) (x-3) graph{2x^2-5x-3 [-6.07, 7.977, -6.18, 0.847]}

Explanation:

Cross factorise the quadratic equation to get the (2$x$+1) ($x$-3) equation.

Then, put:

2$x$+1=0
so make $x$ the subject of the formula
$x$=$- \frac{1}{2}$

And then,
$x$-3=0
so move -3 over
$x$=3

So $- \frac{1}{2}$ and 3 are your $x$- intercepts.

To find the $y$- intercept you need to complete the square, making the quadratic into the form of $a {\left(x - h\right)}^{2}$ + k, so

1. Factorise $2 {x}^{2}$ - 5$x$ with the coefficient of $2 {x}^{2}$ which is 2 (because $x$ has to have a coefficient of only 1 !)
2. Which makes it 2(${x}^{2}$ - 2.5) - 3
3. Then 2 [ ${\left(x - 1.25\right)}^{2}$ - 1.56 ) ] -3
You get 1.25 from dividing 2.5 by 2. You always have to divide the number by 2. Also, you get 1.56 (rounded up to 3 s.f) from squaring 1.25. These are all rules!
4. Then expand the brackets, so
2${\left(x - 1.25\right)}^{2}$-3.12-3
5. 2${\left(x - 1.25\right)}^{2}$- 6.12

So your $y$- intercept is 1.25. (Ignore the negative! Always change it to positive)

And your minimum point will be (1.25,-6.12).