Question

How do you find the vertex and the intercepts for $y = 4{x^2} - 8x + 10$?

Answer 1

We can find the vertex and intercepts of the equation by their general formulas obtained and to find the y-intercept of the given equation, just we need to substitute $x$= 0 in the given equation and solve for y and to find the x-intercept of the given equation, just we need to substitute y = 0 in the given equation and solve for x.

Complete step by step solution:
The given equation is
$y = 4{x^2} - 8x + 10$
The given equation is in the form of $a{x^2} + bx + c$, in which we need to find the vertex of x and y coordinate.
Let us find vertex of x-coordinate
$x = - \dfrac{b}{{2a}}$
From the equation we know that it is quadratic equation and according to it, a = 4 and b = -8
Hence substituting we get
$x = - \dfrac{{ - 8}}{{2\left( 4 \right)}}$
Hence, the vertex of x-coordinate is
$x = \dfrac{8}{8} = 1$
$x = 1$.
Now let us find vertex of y-coordinate
As we obtained the value of x as 1, put this value in given equation to find y-coordinate as
$y = 4{x^2} - 8x + 10$
$y\left( 1 \right) = 4{\left( 1 \right)^2} - 8\left( 1 \right) + 10$
Simplifying the terms, we get
$y\left( 1 \right) = 4 - 8 + 10$
$y\left( 1 \right) = 6$
Therefore, the vertex we obtained is $\left( {1,6} \right)$.
We need to find the x and y intercepts for the equation given, hence to find the x-intercept substitute y=0 and solve for x using quadratic formula
$D = {b^2} - 4ac$
As we know the values of a, b and c, lets substitute them, as
$D = {8^2} - 4\left( 4 \right)\left( {10} \right)$
$D = 64 - 160$
After simplification we get the value of D as
$D = - 96$
Because D < 0, there are no real roots (no x-intercepts). The parabola graph doesn't intersect with the x-axis. The parabola opens upward (a > 0). The parabola graph stays completely above the x-axis.

Note:
As per the given equation consists of x and y terms based on the intercept asked, we need to solve for it. For ex if y-intercept is asked substitute x=0 and solve for y and if x-intercept is asked substitute y=0 and solve for x and the y-intercept of an equation is a point where the graph of the equation intersects the y-axis.