Quadratic and Cubic Graphs

A table of values for $y=x^2$y = x²

The equation $y=x^2$y = x² says: square x to get y. To draw its graph, choose some values of x, work out y for each one, and plot the points.

Take the whole numbers from −3 to 3. Squaring a negative number gives a positive answer, because a negative times a negative is positive: ${\left(-3\right)}^2=\left(-3\right)\times \left(-3\right)=9$(−3)² = (−3) × (−3) = 9. So x = −3 and x = 3 both give y = 9, and x = −2 and x = 2 both give y = 4.

Each row gives one point to plot: x = −3 and y = 9 is the point (−3, 9).

Join the points with a smooth curve

Plot the seven points (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4) and (3, 9), and join them with one smooth curve, not with straight segments.

The rule holds between the plotted points too. At x = 1.5, $y=1.5^2=2.25$y = 1.5² = 2.25. A straight segment from (1, 1) to (2, 4) would pass through (1.5, 2.5), which is above the curve. That is why the points are joined freehand with a curve, not with a ruler.

The seven points from the table, joined by one smooth curve: the graph of $y=x^2$y = x².

The shape of a quadratic graph

The graph of $y=x^2$y = x² is a U-shaped curve called a parabola. It is symmetric about the y-axis, because x and −x have the same square. Its lowest point is (0, 0), where the curve stops falling and starts rising. This point is called the turning point, or the vertex.

An expression whose highest power of x is $x^2$x², such as $x^2-4$x² − 4 or $x^2+3x$x² + 3x, is called a quadratic, and the graph of y equal to any quadratic is a parabola. Take $y=x^2-4$y = x² − 4. Every value of y is 4 less than for $y=x^2$y = x², so the whole U moves 4 down: the turning point is (0, −4). The curve crosses the x-axis where y = 0, at x = 2 and x = −2, because $2^2-4=0$2² − 4 = 0 and ${\left(-2\right)}^2-4=0$(−2)² − 4 = 0.

The graph of $y=x^2-4$y = x² − 4 is the same U, 4 lower. It turns at (0, −4) and crosses the x-axis at −2 and 2.

Cubing keeps the sign

The equation $y=x^3$y = x³ says: multiply three copies of x. This time a negative x gives a negative y: ${\left(-2\right)}^3=\left(-2\right)\times \left(-2\right)\times \left(-2\right)=4\times \left(-2\right)=-8$(−2)³ = (−2) × (−2) × (−2) = 4 × (−2) = −8.

Work out y for x from −2 to 2: −8, −1, 0, 1 and 8. On the right of the y-axis the curve climbs steeply. On the left it falls below the x-axis, because every negative x gives a negative y.

Each row gives one point of $y=x^3$y = x³: x = −2 and y = −8 is the point (−2, −8).

The graph of $y=x^3$y = x³ passes through the origin, rises to the right and falls below the x-axis to the left.

The shape of a cubic graph

The value of y at −x is the negative of the value at x: $2^3=8$2³ = 8 and ${\left(-2\right)}^3=-8$(−2)³ = −8. So the left half of the curve is the right half turned half a turn about the origin. Near the origin the curve flattens out, then it bends and climbs again.

An expression whose highest power of x is $x^3$x³ is called a cubic. The two graphs differ at their ends: the graph of $y=x^2$y = x² rises on both sides of its turning point, while the graph of $y=x^3$y = x³ runs from the bottom left of the grid to the top right.

The usual mistakes

Squaring a negative number and keeping the minus sign. ${\left(-3\right)}^2=\left(-3\right)\times \left(-3\right)=9$(−3)² = (−3) × (−3) = 9, not −9, which is why the parabola comes back up on the left.

Reading the power as a multiplier. $x^2$x² means x × x, not 2x: at x = 4 it is 16, not 8. In the same way, $x^3$x³ at x = 4 is 4 × 4 × 4 = 64, not 12.

Dropping the sign when cubing. ${\left(-2\right)}^3$(−2)³ is −8: the first two factors make 4, and the third makes it negative again.

Joining the points with a ruler, or with a sharp point at the bottom. The graph is a smooth curve, and the bottom of the U is rounded.