Graphical Methods
Algebra · Graphical Methods
Syllabus tag: KCSE | Mathematics | Form 3 | Topic 13 Graphical Methods
Lesson objectives
By the end of this topic, you should be able to:
- Make a table of values from a given relation and draw its graph.
- Draw and interpret graphs of quadratic and cubic functions.
- Solve equations graphically, including by drawing a suitable straight line.
- Determine and interpret the gradient of a curve at a point.
Graphical Methods
A graph turns an equation into a picture, and the picture answers questions the algebra makes hard.
a) Making a table of values
Choose a sensible range of x values, usually given in the question.
Substitute each one, showing the working. A table with the intermediate steps is easier to check.
Watch signs when x is negative. For x = −1 in x² − 2x − 3, the term −2x becomes +2.
b) Drawing the curve
Plot every point carefully.
Join them with a smooth curve, not straight segments. A quadratic curve never has corners.
Extend the curve slightly past the plotted points.
If one point breaks the smooth shape, recheck that row of the table before drawing.
c) Quadratic graphs
A quadratic gives a parabola.
If a is positive, it opens upwards and has a minimum. If negative, it opens downwards and has a maximum.
The roots are where the curve crosses the x-axis, since y is zero there.
The curve is symmetrical about a vertical line through the turning point, which lies midway between the roots.
d) Cubic graphs
A cubic has a distinctive S shape and may cross the x-axis up to three times.
A cubic always crosses at least once, since the curve runs from negative to positive values.
Plot more points than for a quadratic. The shape changes direction twice, and too few points hide that.
e) Solving equations graphically
The roots of f(x) = 0 are where the graph of y = f(x) meets the x-axis.
To solve a different equation from the same curve, rearrange it. One side must be exactly the drawn curve.
Whatever remains on the other side is the line to draw.
The x-coordinates of the crossings are the solutions.
This saves redrawing. One carefully drawn curve can solve several related equations.
f) Gradient of a curve at a point
Unlike a straight line, a curve's gradient changes from point to point.
To find it at a point, draw the tangent there, touching the curve without crossing it.
Then find the gradient of that tangent, as rise over run, using two points far apart on it.
Accuracy depends on the tangent. Use a ruler, judge the touch carefully, and take a large triangle to reduce reading error.
The gradient has meaning. On a distance-time curve it is the speed at that instant. On a velocity-time curve it is the acceleration.
g) Where this is used
Reading solutions where algebra is hard. Finding rates of change from experimental data. Any relationship best understood by its shape.
Words to know
- Table of values -- a list of x values with their corresponding y values.
- Turning point -- a maximum or minimum point on a curve.
- Tangent to a curve -- a straight line touching the curve at one point.
- Ordinate -- the y-value at a given point, used in the trapezium method.
- Linear law -- a relationship converted to a straight line, usually by logarithms.
:::checkpoint Check yourself
- Complete y = x² − 4 for x = −2, 0 and 2.
- What shape is the graph of a quadratic?
- How many times can a cubic cross the x-axis?
- How do you find the gradient of a curve at a point? :::
Bridge to practice
The exercises begin with tables of values and curve sketching, move through graphical solutions and the rearrangement method, and finish with gradients, areas and linear laws. After deriving a line for the rearrangement method, check it contains no x² term before drawing anything.