Graph Of Quadratic Equation
In this chapter, you will learn how to plot a quadratic graph on graph paper step by step. Quadratic equations are commonly tested in exams, and this will be your first introduction to quadratic equations and their graphs. This foundation will prepare you for more advanced applications of quadratic functions in later chapters.
Related Lessons:
Part Of Quadratic Graph
The general equation of a quadratic equation is in the form \(ax^2+bx+c=0\) where \(a\neq0\). The key parts of the quadratic graphs are:
- roots (x-intercept) where \(y=0\)
- y-intercept where \(x=0\)
- turning point
- line of symmetry which is a vertical line that will past through the line of symmetry
Relationship between Line Of Symmetry, Roots and Turning Point
We learn that the general equation of a quadratic equation is in the form \(ax^2+bx+c=0\) where \(a\neq0\).
Let’s say the roots of the quadratic equation are \(\left(x_1, 0\right)\) and \(\left(x_2, 0\right)\).
Equation of Line Of Symmetry is: \(x=\frac{x_1+x_2}{2}\)
To find the y-coordinate of the turning point, we substitute the equation of the Line Of Symmetry into the quadratic equation:
\[
\begin{align*}
y&=ax^2+bx+c\\
&=a\left(\frac{x_1+x_2}{2}\right)^2
+b\left(\frac{x_1+x_2}{2}\right)+c\\
\end{align*}
\]
Therefore, the coordinates of the turning point is:
\(
\left(\frac{x_1+x_2}{2},
a\left(\frac{x_1+x_2}{2}\right)^2
+b\left(\frac{x_1+x_2}{2}\right)
+c\right)
\)
Example 1
The given table of values is for the graph of \(y=x^2-8x+7\).
- Calculate the values of p and of q.
- Taking 2 cm to represent 1 unit on the x-axis, and 1 cm to represent 1 unit on the y-axis, draw the graph of \(y=x^2-8x+7\) for the range \(0\le x\le8\)
- From your graph find
(i) the value of y when \(x=3.5\) (ii)the values of \(x\) when \(y=2\). - State the equation of the line of symmetry of the graph.
Example 2
The graph of \(y=3-2x-x^2\) has been drawn as shown above
- Write down the coordinate of the y-intercept
- Find the coordinates of A and B
Example 3
- Find the values of t when \(h = 2\).
- Given that this firework round explodes when it reaches the greatest possible height, how long will it take for it to explode after it is launched?
- At what height will the firework round explode?