How do you graph y=x^2-14x+24?

Jul 15, 2015

You'll have a parabola passing through:
$x = 0 , y = 24$
$x = 2 , y = 0$
$x = 12 , y = 0$
and vertex at $x = 7 , y = - 25$

Explanation:

This is a Quadratic and its graph will be a Parabola (kind of "U" shaped curve). To plot it you may consider some interesting features of your equation:

1] the coefficient of ${x}^{2}$ is $1 > 0$ so yours will be a "upward" or "happy" parabola (in the shape of a smiling "U");

2] set $x = 0$ into your equation to find the $y$-intercept:
$y = 0 + 0 + 24$
so your parabola will cross the y axis at $y = 24$;

3] set $y = 0$ into your equation to find (if they exists) the $x$-intercepts(s):
${x}^{2} - 14 x + 24 = 0$ solving using the Quadratic Formula you get:
${x}_{1 , 2} = \frac{14 \pm \sqrt{196 - 96}}{2} = = \frac{14 \pm 10}{2} =$
you get two values:
${x}_{1} = \frac{24}{2} = 12$
${x}_{2} = \frac{4}{2} = 2$
so your parabola crosses the x axis at $x = 2 \mathmr{and} x = 12$;

4] the vertex; this is a very important point because it characterize the entire graph setting the central point you need to have when plotting a parabola.
Given the equation in general form $y = a {x}^{2} + b x + c$; in your case you have:
$a = 1$
$b = - 14$
$c = 24$
The coordinates of the vertex are then given as:
${x}_{v} = - \frac{b}{2 a} = - \frac{- 14}{2} = 7$
${y}_{v} = - \frac{\Delta}{4 a} = - \frac{{b}^{2} - 4 a c}{4 a} = - \frac{100}{4} = - 25$

Finally your graph will look like:
graph{x^2-14x+24 [-83.3, 83.3, -41.64, 41.64]}