Question

How to make the parabola of $y = {x^2} - 4x$.

Answer 1

Hint- In order to draw the graph of given parabola equation first we have to find the vertex of parabola which can be calculated by using the formula as vertex $(aos,f(aos))$ so we will calculate $aos$ in the solution further we will find intercept points of given parabola and by getting it we will plot the graph.

Complete step-by-step solution -
Given equation of parabola is $y = {x^2} - 4x$
At first we will calculate its vertex
As we know that if parabola equation is given as
$y = a{x^2} + bx + c$
Then axis of symmetry can be calculated by
$x = \dfrac{{ - b}}{{2a}}$
And vertex of given parabola equation is given as
Vertex $(aos,f(aos))$
Where $c{\text{ }} = {\text{ }}y$ intercept
But here our function is
$y = {x^2} - 4x$
By comparing the given equation with general equation we have
$a = 1$
$b = - 4$
and $c = 0$
Therefore , axis of symmetry =
$x = \dfrac{{ - b}}{{2a}} = \dfrac{{ - ( - 4)}}{{2 \times 1}} = \dfrac{4}{2} = 2$
$f(aos)$ means we put the $aos$ back in our function as $x$ and solve for $y$:
f(aos) = {(2)^2} - 4(2) \\\ f(aos) = 4 - 8 \\\ f(aos) = - 4 \\\
Now vertex will be expressed as
Vertex = $(2, - 4)$
Now we will proceed further by calculating the intercept points of $x$ axis and $y$ axis
For $y$ intercept we have to put the value of $x$ as 0
So , at $x = 0$,
\because y = {x^2} - 4x \\\ y = {(0)^2} - 4(0) \\\ y = 0 \\\
$y$ intercept $(0,0)$
Similarly , For $x$ intercept we have to put the value of $y$ as 0
So, at $y = 0$ ,

0 = {x^2} - 4x \\\ {x^2} - 4x = 0 \\\ x(x - 4) = 0 \\\ x = 0{\text{ and }}x = 4 \\\ $$ By simplifying above equation we get two values of $$x$$ as $$x{\text{ }} = {\text{ }}0{\text{ }} and {\text{ }}x{\text{ }} = {\text{ }}4$$ Therefore, $$x$$ intercept are $(0,0)$ and $(4,0)$ Now with the help of obtained data we will draw parabola of given equation Vertex = $(2, - 4)$ $$x$$ intercepts are $(0,0)$ and $(4,0)$ $y$ intercept $(0,0)$  Note- The axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves. The x-coordinate of the vertex is the equation of the axis of symmetry of the parabola. For a quadratic function in standard form, $y = a{x^2} + bx + c$ , the axis of symmetry is a vertical line $x = \dfrac{{ - b}}{{2a}}$ .