Graphing Quadratic Equations
In this section we shall investigate a quadratic curve by manually calculating several coordinates on the curve. The plotted points will trace the quadratic curve. We shall learn that even though individual quadratic curves are unique, they all have some features in common.
A Common Misunderstanding
Please note the following to prevent a common misunderstanding:
The same y-value for x=3 and x=-3, since squaring involves the product of two negative numbers which is positive. The squaring operation does not operate on the "a" coefficient which in this case is a=-2.
Exercise: Plot Points on a Quadratic Curve
Substitute the given x value for the quadratic equation and calculate the y-value. The coordinate (x,y) is plotted automatically. Be prepared with pen and paper and repeat the exercise a few times; the hard work will be worth it. In later sections quadratic equation will be explored in depth; but for now it would extremely useful if you repeat the exercise a few times.
The thin blue line is the smooth curve that has been drawn though the points by the computer, in similar test questions or homework exercises you must draw a smooth line by hand.
Features of a Quadratic Graph
A computer is much faster than a human being at performing simple repetitive calculations. A computer graphing app rapidly samples many closely spaced x values and calculates the corresponding y values, the results are shown below:
The curves are different for different quadratic equations, but they all have some features in common:
Solutions
The solutions are the x values for which y=0. The solutions can be found by examining the graph.Axis of Symmetry
The axis of symmetry (black line) is a vertical line that passes through the midpoint between the two solutions. The axis of symmetry is a mirror symetry line. You can place a mirror on the line and observe the reflection of the curve. The moving points image under reflection is shown also.Vertex
The vertex is the minimum or maximum point and it is the intersection between the axis of symmetry and the quadratic curve. A quadratic curve with a minimum point is said to be concave up and one with a maximum point is said to be concave down.The y intercept
The y intercept can be found by substituting x=0 into the quadratic equation. The intercept coordinate is (0,c)
Exercise: Quadratic curves
Use the graph to determine the solutions, the axis of symmetry, and the vertex