Question

How do you find the important points to graph $y = {x^2} - 2x - 3$ ?

Answer 1

Hint:We know that the given equation $y = {x^2} - 2x - 3$is the equation of a parabola. Now, for the graph of a parabola, we need three main points. These three important points are y intercept, x intercept and vertex point.

Complete step by step answer:
We will first find the y intercept of the graph. For this we will take $x = 0$.
$y = {x^2} - 2x - 3 \\
\Rightarrow y = - 3 $
Therefore, the y intercept of the graph is $\left( {0, - 3} \right)$. Now, we will find the x intercept of the graph. For this we will take $y = 0$.
$y = {x^2} - 2x - 3 \\
\Rightarrow 0 = {x^2} - 2x - 3 $
For solving this, we will factor the polynomial ${x^2} - 2x - 3$by splitting the middle term.
Therefore, we can write
\[{x^2} - 3x + x - 3 = 0 \\
\Rightarrow x\left( {x - 3} \right) + 1\left( {x - 3} \right) = 0 \\
\Rightarrow \left( {x - 3} \right)\left( {x + 1} \right) = 0 \\ \]
$\Rightarrow x - 3 = 0 \\
\Rightarrow x = 3 \\ $
$\Rightarrow x + 1 = 0 \\
\Rightarrow x = - 1 \\ $
Thus, we have two x intercepts for the given graph which are $\left( {3,0} \right)$and $\left( { - 1,0} \right)$. We will now find the vertex of the parabola. For this we will use the completing the square method and factorize the right hand side of the equation.
$y = {x^2} - 2x + 1 - 3 - 1 \\
\therefore y = {\left( {x - 1} \right)^2} - 4 \\ $
If we compare this equation with the standard equation of parabola $y = {\left( {x - h} \right)^2} + k$, where, $\left( {h,k} \right)$ is the vertex point of the parabola, therefore in this case the vertex point is $\left( {1, - 4} \right)$.

Thus, the important points of graph $y = {x^2} - 2x - 3$are $\left( {0, - 3} \right)$, $\left( {3,0} \right)$, $\left( { - 1,0} \right)$ and $\left( {1, - 4} \right)$.

Note:In this type of question, there are three steps to find the important points of the graph of parabola. In the first step, we determine the y intercept of the graph by putting the x coordinate as zero. In the second step, we determine the x intercept of the graph by putting the y coordinate as zero. After that, finally we determine the vertex point by making the equation in the form of the standard equation of parabola.