# How do you graph f(x)=3x^2 + x - 5?

Jun 13, 2015

Plot points, using a graphing utility, or manipulate the form of the equation so that it's elements reveal the graph's behaviors.

#### Explanation:

Plotting a few points is an easy method. Simply plug in some values for $x$ that are negative, some that are close to zero, and some that are positive.

When you plot these points, you get following. Note that the red values beneath the dots are in the form $\left(x , f \left(x\right)\right)$:

Drawing a smooth line through them gives you the graph you are looking for. Using your graphing utility, like a Ti-83, is even better. Finally, you can manipulate the equation to give to information about the behavior of the parabola. This is known as graphing a quadratic function using transformations.

First, complete the square on the right side:

$f \left(x\right) = 3 {x}^{2} + x - 5$
$= 3 \left({x}^{2} + \frac{1}{3} x\right) - 5$ factor out a three
$= 3 \left({x}^{2} + \frac{1}{3} x + \frac{1}{36}\right) - 5 - \frac{3}{36}$ completing the square
$= 3 {\left(x + \frac{1}{6}\right)}^{2} - \frac{59}{12}$
$\cong 3 {\left(x + 0.167\right)}^{2} - 4.92$

The $3$ in front means you should start with the stretched parabola, $f \left(x\right) = 3 {x}^{2}$. The $\frac{1}{6}$ means shift the parabola horizontally to the left by one-sixth. Finally, the $- 4.92$ tells you to drop the parabola vertically down about $5$ units. Finis!