Harder Graphs

Understanding Quadratic Graphs

Quadratic graphs are shaped like a curve or a ‘U’ shape, called a parabola.
The equation of a quadratic graph is given in the form of y = ax² + bx + c.
A positive ‘a’ value means the graph opens upwards, a negative ‘a’ value means it opens downwards.
The vertex of the graph is the maximum or minimum value of ‘y’. At the vertex, x = -b/2a.
The line of symmetry is a vertical line through the vertex.
Quadratic graphs can intersect the x-axis at two points, one point, or doesn’t intersect at all. These intersection points are the roots or zeroes of the quadratic equation.
The discriminant (b² - 4ac) helps predict the number of roots a quadratic equation has. If discriminant > 0, there are two roots. If it equals to 0, there is one root. If it is less than 0, there are no real roots.

To plot a quadratic graph, start with the y-intercept which corresponds to the ‘c’ value in the equation.
The next point of interest is the vertex of the parabola. Use the formula for the vertex coordinates to plot this point.
Once the y-intercept and vertex are plotted, additional points may be plotted using a table of values.
When several points are connected, the classic parabolic curve forms.

Cubic graphs are represented by functions of the form y = ax³ + bx² + cx + d.
They change direction once or twice forming up to three turning points.
Cubic functions could pass through x-axis at three, two, one or no x-intercept based on the roots of cubic equation.

To plot a cubic graph, identify a number of key features: y-intercept, x-intercepts or turning points.
The y-intercept can be found by setting x to zero in the function.
Determine x-intercepts or roots by solving the cubic equation.
An efficient way to draw cubic graphs is to construct a table of values to determine extra points to plot.
After plotting, draw a smooth curve and run it through the essential points.

To graph a circle, identify its centre (h, k) and radius ‘r’.
Plot the centre, and from the centre plot additional points using the radius in every cardinal direction.
Connect these points to form the circle.