2.6 Graphical Methods

• Graphs of quadratic equations and graphical solution of a linear and a quadratic equation
• Graphs of cubic relations
• Average rate of change and rate of change at an instant
• Empirical data, their graphs and the line of best fit
• Reduction of non-linear relations to linear form
• Equation of a circle; conic curves: circle and parabola

• solve quadratic and cubic equations graphically
• solve simultaneous equations of a linear and a quadratic or cubic equation graphically
• determine average rate of change and rate of change at an instant from a graph
• draw graphs of empirical data and line of best fit in different situations
• convert nonlinear equations to linear form in different situations
• determine constants in nonlinear equations in different situations
• determine the centre, radius and equation of a circle in different situations
• draw graphs of circles and parabolas in different situations
• promote the use of graphical solutions in day to day life

“How are relations presented in real life?”

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1. The graph of a quadratic
   40 min · 1 practice tasks · quiz
   Plot y = x^2 - x - 6, mark the intercepts, and name the turning point and the axis of symmetry.
2. Roots from the x-axis
   40 min · 1 practice tasks · quiz
   Read two roots, one touching root, or no real root from the way a parabola meets the x-axis.
3. A horizontal line on a parabola
   40 min · 1 practice tasks · quiz
   Use one quadratic graph to solve a related equation by reading where the curve meets a chosen horizontal line.
4. A line and a quadratic
   40 min · 1 practice tasks · quiz
   The meeting points of a parabola and a straight line are the solutions of the simultaneous equations.
5. The graph of a cubic
   40 min · 1 practice tasks · quiz
   Plot y = (x - 1)(x - 2)(x - 3), including the points between the roots so the bends are visible.
6. Solving a cubic from its graph
   40 min · 1 practice tasks · quiz
   Read three roots on the x-axis, then one solution where the same cubic meets the line y = 6.
7. A line and a cubic
   40 min · 1 practice tasks · quiz
   y = x^3 and y = x meet at three points, while y = 8 meets the cubic once.
8. Average rate from a chord
   40 min · 1 practice tasks · quiz
   The gradient of the chord between two points of a graph is the average rate of change on that interval.
9. Rate from a drawn tangent
   40 min · 1 practice tasks · quiz
   The gradient of the tangent at one point is the instantaneous rate, read from two points on that straight tangent.
10. A line of best fit
    40 min · 1 practice tasks · quiz
    Plot classroom prices, draw the straight line through the agreeing points, and read its gradient and intercept.
11. When a straight fit fails
    40 min · 1 practice tasks · quiz
    A value past the plotted points is an extension, and square areas bend away from the straight line through their ends.
12. Plotting y against x squared
    40 min · 1 practice tasks · quiz
    The law y = ax^2 + b is a straight line when y is plotted against x^2, and the gradient and intercept are the constants.
13. Plotting y against 1/x
    40 min · 1 practice tasks · quiz
    The law y = a/x + b is a straight line of gradient a when y is plotted against 1/x.
14. A power law on log axes
    40 min · 1 practice tasks · quiz
    For y = ax^n, the graph of log y against log x is straight, with gradient n and intercept log a.
15. Centre, radius and equation
    40 min · 1 practice tasks · quiz
    Write a circle from its centre and radius, and recover the centre and radius by completing the square.
16. Drawing a circle and a parabola
    40 min · 1 practice tasks · quiz
    Sketch the circle from its centre and radius, and sketch the sideways parabola y^2 = 8x from a table.