Quadratic Equations
Graph of y = x2
Properties of the quadratic curve:
1. There is one turning point, which is either maximum or minimum.
2. There are no asymptotes; the function is defined for all values of x.
3. The curve is symmetrical about the turning point.
The quadratic curve can be sketched from the technique described earlier. However, there are 2 simpler methods.
Completing the Square
The completed square form of the quadratic equation is
y = a (x - h)2 + k
Information about the quadratic curve can be found from this form:
1. If a>0, the curve has a minimum turning point. Conversely, if a<0, the curve has a maximum turning point.
2. The x-coordinate of the turning point is given by h.
3. The y-coordinate of the turning point is given by k.
4. X- and y-intercepts has to be calculated the usual way, that is, set y=0 and x=0 respectively.
Factorized Form
The general factorized form of the quadratic equation is
y = a (x - c) (x - d)
The information derived from this form is slightly different from those above:
1. C and d are the x-intercepts.
2. Y-intercept is calculated by setting x=0.
3. If a>0, the curve has a minimum turning point. Conversely, if a<0, the curve has a maximum turning point.
4. The x-coordinate of the turning point is calculated by taking the average of c and d. Recall Property 3 above. The y-coordinate is calculated by substituting the x-coordinate into the equation.
Which method is better? This depends on what kind of information is given. However, if asked to simple sketch a quadratic expression, both methods work just as well. Review the example below and see which you prefer.
Example: Sketch the graph of y = x2 - 2x - 8.
Solution:
Method 1: Completing the square.
y = x2 - 2x - 8
= (x - 1)2 - 1 - 8
= (x - 1)2 - 9
Turning point = (1, -9)
Y-intercept = (0, -8)
X-intercepts: Solve x2 - 2x - 8 = 0 --> (4, 0) and (-2, 0)
Method 2: Factorization
y = x2 - 2x - 8
= (x - 4)(x + 2)
X-intercepts = (4, 0), (-2, 0)
Y-intercept = (0, -8)
Turning point: x-coordinate = (4 - 2)/2 = 1
y-coordinate = -9 (subst into equation)
Higher Degree Equations
Graph of y = x3
Properties of the cubic curve:
1. There is one stationary point, which is a point of inflexion.
Some cubic curves may have two turning points instead, depending on the equation. There may be one, two or three x-intercepts.
2. There are no asymptotes; the function is defined for all values of x.
3. The curve is symmetrical about origin.
Example: Sketch the graph of y = x3 - x.
Solution:
X-intercepts: when y = 0,
x (x + 1)(x - 1) = 0 --> x = 0, x = -1, x = 1
Y-intercept = (0, 0)
Turning point:
dy/dx = 3x2 - 1 = 0
x = ± 1/Ö3
when
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Curve Sketching | Quadratic Equation | Rectangular Hyperbolas | Conic sections | Modulus Functions
