Plotting quadratic and cubic graphs – Worked Examples Mastering Mathematics © Hodder and Stoughton 2014 Toolbox Drawing a graph Solving an equation with a graph Intersecting graphs Contents

Plotting quadratic and cubic graphs – Worked Examples Mastering Mathematics © Hodder and Stoughton 2014 A cubic equation has x 3 as the highest power of x. A negative number squared becomes positive. (-4) 2 = 16 A negative number cubed remains negative. (-2) 3 = -8 A quadratic equation has x 2 as the highest power of x. Toolbox MenuBackForward To draw the graph of a quadratic or cubic equation, create a table: x x2x x y = x 2 – 3x y x y x

Plotting quadratic and cubic graphs – Worked Examples Mastering Mathematics © Hodder and Stoughton 2014 x 3 is negative when x is negative. +4 is always 4, as it is independent of x. 1.Complete the table below. 2.Draw the graph of y = x Drawing a graph MenuToolboxBackForwardAnswer 1 Answer 2 Draw the graph of y = x on the same axes. What do the two graphs have in common? More x x3x y = x x x3x y = x The red graph shows y = x y x

Plotting quadratic and cubic graphs – Worked Examples Mastering Mathematics © Hodder and Stoughton 2014 x 2 is never negative. 1.Complete the table below. 2.Draw the graph of y = x 2 – 3x + 2. Solving an equation with a graph MenuToolboxBackForwardCont/dAnswer 1 Answer 2 x x2x2 -3x +2 x 2 – 3x + 2 x x2x x x 2 – 3x The red graph shows y = x 2 – 3x + 2. y x

Plotting quadratic and cubic graphs – Worked Examples Mastering Mathematics © Hodder and Stoughton Use the graph to find the solutions to the equation x 2 – 3x + 2 = 5. 2.Check your solutions in the equation and explain your answers. MenuToolboxBackForwardAnswer 1 Answer 2 More y x Solving an equation with a graph Use your answer to question 2 to answer these. Is the exact positive solution more or less than x = 3.8? Is the exact negative solution more or less than x = -0.8? How do you know? When x = -0.8, x 2 – 3x + 2 = = 5.04 When x = 3.8, x 2 – 3x + 2 = – = 5.04 The answers are very close to 5. The solutions are correct to 1 decimal place. The solutions to the equation x 2 – 3x + 2 = 5 are the intersections of the graphs y = x 2 – 3x + 2 and y = 5. The solutions are x = -0.8 and x = 3.8. y x

Plotting quadratic and cubic graphs – Worked Examples Mastering Mathematics © Hodder and Stoughton 2014 The curve shown has equation y = x 3 – 2x Write down the equation of the straight line. 2.Write down the x coordinates of the points of intersection of the straight line and the curve. Intersecting graphs They intersect at x = -2.2, x = 0.5 and x = 1.7. MenuToolboxBackForwardCont/dAnswer 1 Answer 2 y x The y intercept is at (0, -1). The line also passes through (2, 3) so the gradient is: The equation is y = 2x – 1. 3 – -1 2 – = = 2

Plotting quadratic and cubic graphs – Worked Examples Mastering Mathematics © Hodder and Stoughton 2014 The graph shows the curve y = x 3 – 2x + 1 and the straight line y = 2x – 1. The lines intersect when x = -2.2, x = 0.5 and x = 1.7 The intersections are solutions to the equations y = x 3 – 2x + 1 and y = 2x – 1. They are solutions to the equation: x 3 – 2x + 1 = 2x – 1 1.x = -2.2, x = 0.5 and x = 1.7 are solutions to an equation. Write down the equation. 2.Rewrite the equation so the right-hand side is equal to 0. Intersecting graphs MenuToolboxBackAnswer 1 Answer 2 More y x Check that x = -2.2, x = 0.5 and x = 1.7 are solutions to x 3 – 4x + 2 = 0. Remember the solutions are to 1 decimal place so the equation should be very close to but not exactly 0. x 3 – 2x + 1 = 2x – 1 Subtract 2x from both sides:x 3 – 4x + 1 = -1 Add 1 to both sides:x 3 – 4x + 2 = 0

Plotting quadratic and cubic graphs – Worked Examples Mastering Mathematics © Hodder and Stoughton 2014 Editable Teacher Template MenuToolboxBackForwardAnswer 1More Information 1.Task – fixed 2.Task – appears Answer 1 Answer 2 More Answer 2