Graphing Functions
Draw the Graph of a Quadratic Function
Draw the graph of the function
\(f(x)=x^{2}-x-2\)
in the domain \(-2\le x\le 3,\;x\in\mathbb{R}\).
\(f(x)=x^{2}-x-2\)
in the domain \(-2\le x\le 3,\;x\in\mathbb{R}\).
Step 1: Table of values
Use \(f(x)=x^{2}-x-2\) to find the corresponding
\(y\)-values for integer \(x\) from \(-2\) to \(3\).
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|---|---|
| \(f(x)\) | \(4\) | \(0\) | \(-2\) | \(-2\) | \(0\) | \(4\) |
So the points are \((-2,4),(-1,0),(0,-2),(1,-2),(2,0),(3,4)\).
Step 2: Plot and join smoothly
Plot all six points on the co-ordinate plane.
Join the points with a smooth U-shaped curve (a parabola) opening
upwards. The lowest point lies midway between the roots, at
\(x=\tfrac{1}{2}\), giving the vertex
\(\left(\tfrac12,-\tfrac{9}{4}\right)\).
Finish the curve at \(x=-2\) and \(x=3\) to match the domain
\(-2\le x\le3\).
Draw the Graph of a Quadratic Function
Draw the graph of the function
\(f(x)=2x^{2}+3x-2\)
in the domain \(-3\le x\le 2,\;x\in\mathbb{R}\).
\(f(x)=2x^{2}+3x-2\)
in the domain \(-3\le x\le 2,\;x\in\mathbb{R}\).
Step 1: Table of values
Use \(f(x)=2x^{2}+3x-2\) to find the corresponding
\(y\)-values for integer \(x\) from \(-3\) to \(2\).
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|---|---|
| \(f(x)\) | \(7\) | \(0\) | \(-3\) | \(-2\) | \(3\) | \(12\) |
So the points are \((-3,7),(-2,0),(-1,-3),(0,-2),(1,3),(2,12)\).
Step 2: Plot and join smoothly
Plot all six points on the co-ordinate plane.
Join the points with a smooth U-shaped curve (a parabola) opening
upwards. The lowest point lies at the vertex
\(x=-\dfrac{b}{2a}=-\dfrac{3}{4}\), giving
\(\displaystyle f\!\left(-\dfrac{3}{4}\right)=-\dfrac{25}{8}\).
Finish the curve at \(x=-3\) and \(x=2\) to match the domain
\(-3\le x\le2\).
