Steps Squares and cubes Quadratic graphs Cubic graphs Solving equations

Squares and cubes Squaring means multiplying a number by itself. 72 = 7 × 7 = 49 Cubing a number means multiplying three of the numbers together. 43 = 4 × 4 × 4 = 64 24 means 2×2×2×2. 25 means 2×2×2×2×2. Find the value of 24, (-2)4, 25 and (-2)5. You have already met square numbers and cube numbers. 32, or 3 squared, means 3 × 3 = 9. There are 9 squares in the picture. 23, or 2 cubed, means 2 × 2 × 2 = 8. There are 8 cubes in the picture. You are going to draw graphs with terms in x2 and x3. To do this you will need to use negative (-) values of x as well as positive (+). 1. What is the value of (-8)2? 64  -64  2. What is the value of (-2)3? 8  -8  Opinion  is correct. (-2)3 = (-2) × (-2) × (-2) = 4 × (-2) = -8 In this case you are multiplying three negative numbers and so the answer is negative. Opinion  has a positive answer so it is wrong. Opinion  is correct. (-8)2 = (-8) × (-8) = +64 Remember multiplying two negatives makes a positive. Opinion  has a negative answer and so it is wrong. Menu Back Forward More Vocabulary Q 1 Opinion 1 Opinion 2 Answer Q 2 Opinion 1 Opinion 2 Answer

Quadratic graphs x Area = x2 x A quadratic graph is one where the highest power of x is x2. Any other terms are either in x or a constant (a number). Many equations do not produce straight lines. Here is a table showing values of x and y when y = x2 Complete the table for y = x2 + x. Will y have any negative values? x Area = x2 x x -3 -2 -1 1 2 3 x2 9 4 y = x2 + x 6 12 x -3 -2 -1 1 2 3 y = x2 9 4 1. How can you tell from the table that the graph of y = x2 is not a straight line? 2. Does y = x2 ever have any negative values of y? When the values of x increase in equal steps, the values of y do not.  The values of y decrease at first, and then increase.  No, a square is always positive or zero.  Yes, when x is between -1 and 0.  Both Opinions are correct. They are saying the same thing in different ways. Opinion  is more precise. No. Opinion  is incorrect, squaring a negative number always results in a positive number. Menu Back Forward Cont/d More Vocabulary Q 1 Opinion 1 Opinion 2 Answer Q 2 Opinion 1 Opinion 2 Answer

Quadratic graphs y x The graph shows y = x2. It is symmetrical. 1. Where is the line of symmetry of y = x2? 2. Here is the table for y = x2 + 3x – 3 Draw the graph of y = x2 + 3x – 3. x -3 -2 -1 1 2 3 x2 9 4 +3x -9 -6 6 y = x2 +x - 3 -5 7 15 x The purple curve.  y = 0 The blue curve.  The y-axis. The graphs of y = x2 and y = x2 + 3x – 3 are identical in shape. Draw the graph of y = x2 – x – 2. Is it the same shape? The correct answer is the purple curve Opinion  incorrectly assumes the curve is symmetrical about the y-axis. It is the y axis. y = 0 is the equation of the x-axis. Menu Back Forward Cont/d More Q 1 Opinion 1 Opinion 2 Answer Q 2 Opinion 1 Opinion 2 Answer

Quadratic graphs y x For this table you have to work out the value of -x2.So when x is negative there are three minuses, making a minus overall. Opinion  made the mistake of making the -x2 terms positive. Opinion  is correct. All quadratic graphs have the same U-shape. Look at the equation y = 2x – x2. The x2 term is negative. Complete the table for y = 2x – x2. x -2 -1 1 2 3 2x -4 -x2 y = 2x – x2 x -2 -1 1 2 3 2x -4 4 6 -x2 -9 y = 2x – x2 -8 -3 2. Here is the graph of y = 2x – x2. What can you say about its shape? It is the usual U-shape but has been turned upside down. This is because the x2 term is negative instead of positive. Draw the graph of y = x2 – 2x. Where does this curve cross y = 2x – x2? It is the same shape as all the quadratic curves, but it is upside down.  It is the same shape as all the quadratic curves.  x -2 -1 1 2 3 2x -4 4 6 -x2 9 y = 2x – x2 8 12 x -2 -1 1 2 3 2x -4 4 6 -x2 9 y = 2x – x2 -8 -3 8 15 Menu Back Forward More Q 1 Opinion 1 Opinion 2 Answer Q 2 Opinion 1 Opinion 2 Answer

Cubic graphs y x Cubic graphs have their own distinctive shape. 1. Compete this table of values for y = x3. x -3 -2 -1 1 2 3 y = x3 27 x 2. Describe the gradient of the graph  It decreases at first and then increases. x -3 -2 -1 1 2 3 y = x3 27 8 It is always increasing. Remember that when x is negative, x3 is negative. Opinion  makes the mistake of saying it is positive. x -3 -2 -1 1 2 3 y = x3 -27 -8 8 27  x -3 -2 -1 1 2 3 y = x3 -27 -8 8 27 What will the graph of y = -x3 look like? A cubic equation has x3 as its highest power of x. The gradient of the graph is always positive, except at x = 0 where it is 0. It decreases until x = 0 and then increases. Menu Back Forward Cont/d More Vocabulary Q 1 Opinion 1 Opinion 2 Answer Q 2 Opinion 1 Opinion 2 Answer

Cubic graphs The green curve crosses the x-axis at x = -2. So substituting x = -2 in the right curve will give y = 0. A y = x3 + 4x = -8 + -8 = -16 B y = x3 - 4x = -8 + 8 = 0 C y = x3 + 4 = -8 + 4 = -4 So Opinion  is correct. The green curve is B, y = x3 - 4x. Opinion  mistakenly states that the cube of a negative number is positive. The red curve intercepts the y-axis at y = 4. This is where x = 0. Substituting x = 0 gives A y = x3 + 4x = 0 + 0 = 0 B y = x3 – 4x = 0 – 0 = 0 C y = x3 + 4 = 0 + 4 = 4 So the red curve must be C, y = x3 + 4. Opinion  makes a mistake with curve A. It states 4 × 0 = 4. y Look at the graph. There are three curves: A: y = x3 + 4x B: y = x3 – 4x C: y = x3 + 4 For each equation, find the value of y when x = 0. What is the equation of the red curve? x 2. For each equation, find the value of y when x = -2. What is the equation of the green curve? A y = x3 + 4x = 0 + 4 = 4 B y = x3 - 4x = 0 – 4 = -4 C y = x3 + 4 = 0 + 4 = 4 The red curve could be y = x3 + 4x or y = x3 + 4.  A y = x3 + 4x = -8 + -8 = -16 B y = x3 - 4x = -8 + 8 = 0 C y = x3 + 4 = -8 + 4 = -4 The green curve is y = x3 - 4x.  These three curves have similar shapes but an important difference. What do you notice about the gradients of each curve when x = 0? A y = x3 + 4x = 0 + 0 = 0 B y = x3 - 4x = 0 – 0 = 0 C y = x3 + 4 = 0 + 4 = 4 The red curve must be y = x3 + 4.  A y = x3 + 4x = 8 + -8 = 0 B y = x3 - 4x = 8 + 8 = 16 C y = x3 + 4 = 8 + 4 = 12 The green curve is y = x3 + 4x.  Menu Back Forward More Q 1 Opinion 1 Opinion 2 Answer Q 2 Opinion 1 Opinion 2 Answer

Solving equations y x You can use a graph to solve an equation. Here is the graph of y = x2 – 4. 1. What are the coordinates of the intercepts on the x-axis? 2. What does this tell you about the value of y when x = -2? (-2, 0) and (2, 0)  The intercepts are (-2, 0) and (2, 0). Opinion  is correct. Opinion  has the intercept on the y-axis. (0, -4)  Use the graph to solve the equation x2 – 3 = 1. When x = -2, y does not exist. When x = -2, y = 0. Also, when x = 2, y = 0. So the solution of the equation x2 – 4 = 0 is x = -2 or x = 2. When x = -2, y = 0 Menu Back Forward Cont/d More Q 1 Opinion 1 Opinion 2 Answer Q 2 Opinion 1 Opinion 2 Answer

Solving equations y x You can use one graph to solve many equations. Here is the graph of y = x3 – 2x. 1. Write the equation x3 – 2x – 5 = 0 in the form x3 – 2x = ... x 2. How can you use the graph to solve the equation x3 – 2x – 5 = 0? Subtract 5 from each side: x3 – 2x = -5  Draw the graph of y = x3 – 2x – 5 and see where the curve crosses the x-axis.  Add 5 to each side: x3 – 2x = 5  x3 – 2x – 5 = 0 is the same as x3 – 2x = 5. Draw the graph of y = 5 and find the value of x where it crosses the curve. That is where y = x3 – 2x and y = 5 have the same value and so is the solution of the equation. It is x = 2.1 (to 1 decimal place). Opinion  said to draw y = x3 – 2x – 5 and see where it crosses the x-axis. You could do this but it would take you much longer. Opinion  is correct: x3 – 2x = 5. You must add 5 to each side to eliminate the -5. Opinion  is wrong. Subtracting 5 makes the left hand side into: x3 – 2x – 10 and not: x3 – 2x. How could you use the graph to solve the equation y = x3 – 3x? Draw the graph of y = 5 and see where it crosses y = x3 – 2x.  Menu Back More Q 1 Opinion 1 Opinion 2 Answer Q 2 Opinion 1 Opinion 2 Answer

Editable Teacher Template Information Vocabulary 1. Task – fixed More 2. Task – appears Q1 Opinion 1 Q1 Opinion 2 Q1 Answer Q2 Opinion 1 Q2 Opinion 2 Q2 Answer Menu Back Forward More Vocabulary Q 1 Opinion 1 Opinion 2 Answer Q 2 Opinion 1 Opinion 2 Answer