# What are the vertex, axis of symmetry, maximum or minimum value, domain, and range of the function, and x and y intercepts for  y=x^2-10x+2?

Apr 29, 2015
• $y = {x}^{2} - 10 x + 2$ is the equation of a parabola which will open upwards(because of the positive coefficient of ${x}^{2}$)
So it will have a Minimum

• The Slope of this parabola is
$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 x - 10$
and this slope is equal to zero at the vertex
$2 x - 10 = 0$
$\to 2 x = 10 \to x = 5$

• The X coordinate of the vertex will be $5$

$y = {5}^{2} - 10 \left(5\right) + 2 = 25 - 50 + 2 = - 23$
The vertex is at color(blue)((5,-23)

and has a Minimum Value color(blue)(-23 at this point.

• The axis of symmetry is color(blue)(x=5

• The domain will be color(blue)(inRR(all real numbers)

• The range of this equation is color(blue)({y in RR : y>=-23}

• To get the x intercepts, we substitute y = 0
${x}^{2} - 10 x + 2 = 0$
We get two x intercepts as color(blue)((5+sqrt23) and (5-sqrt23)

• To get the Y intercepts, we substitute x = 0
$y = {0}^{2} - 10 \cdot 0 + 2 = 2$
We get the Y intercept as color(blue)(2

• This is how the Graph will look:
graph{x^2-10x+2 [-52.03, 52.03, -26, 26]}