Quadratic functions, equations and inequalities · Year 10
In Quadratic functions, equations and inequalities (Year 10) you practise 6 skills: Working out values of a quadratic function · The turning point: minimum or maximum · Reading the graph of a quadratic · Factorised form and where the graph crosses the axis · Matching a rule to a graph · Problems that lead to a quadratic equation. A quadratic function has a squared term in its rule, and its graph is a parabola. When the coefficient of the squared term is positive, the parabola opens upwards and has a minimum point; when it is negative, it opens downwards and has a maximum point. To work out a value, put the number in place of x, and write a negative number in brackets.
When the rule is a number times a bracket squared, plus a number, you can read the turning point from it: the number inside the bracket, with its sign changed, is the first coordinate, and the number outside the bracket is the second. A minus inside the bracket moves the parabola right, and a plus moves it left. When the rule is a product of two brackets, it shows where the graph crosses the horizontal axis: each bracket is 0 at one point. The line of symmetry runs halfway between them, through the turning point. From the graph you can read the turning point, where it crosses the axes, where the function is increasing or decreasing, and where it is positive or negative. Solve a quadratic equation with the quadratic formula, after rearranging it so that one side is 0. First work out the expression under the square root: when it is positive there are two solutions, when it is 0 there is one, and when it is negative there are none. In a word problem, check at the end which solution fits: a length or a positive number cannot come out negative. To solve a quadratic inequality, use the parabola: find where the expression is 0, and check which way it opens. With ≥ or ≤, the crossing points belong to the solution too.
Tip: Before you choose a rule for a graph, check which way the parabola opens and where its turning point is. Those two things rule out nearly every wrong rule.
What you learn
Working out values of a quadratic function
For example: The function is f(x) = −3x² . Work out f(5) .
Answer and solution
The answer: −75
f(5) = (−3) × 5² = (−3) × 25 = −75
The turning point: minimum or maximum
For example: What are the coordinates of the turning point of the parabola y = 3x² − 7 ?
Answer and solution
The answer: (0, −7)
There is no bracket, so the turning point is on the vertical axis: its first coordinate is 0. The number outside the bracket is −7, so the turning point is (0, −7) .
Reading the graph of a quadratic
For example:
The picture shows the graph of a quadratic function. What are the coordinates of the vertex of the parabola?Answer and solution
The answer: (−4, −2)
(−4, −2)
Factorised form and where the graph crosses the axis
For example: The function is f(x) = (x + 7)(x − 9) . Where does its graph cross the horizontal axis? Write the values of x.
Answer and solution
The answer: x = −7 or x = 9
x + 7 = 0 x = −7 x − 9 = 0 x = 9
Matching a rule to a graph
For example:
Which rule fits the graph?- y = −2x² − 5
- y = 2x² + 5
- y = −2x² + 5
- y = 2x² − 5
Answer and solution
The answer: y = −2x² + 5
The parabola opens downwards, and its turning point is (0, 5) . The rule that fits: y = −2x² + 5
Problems that lead to a quadratic equation
For example: A rectangle is 8 cm longer than it is wide, and its area is 33 cm². How wide is it?
Answer and solution
The answer: 3 cm
Call the width x. The length is x + 8, and the area is length times width: x(x + 8) = 33 x² + 8x − 33 = 0 The expression under the square root in the quadratic formula: 8² − 4 × 1 × (−33) = 196 √196 = 14 x = (−8 + 14) ÷ 2 = 3 x = (−8 − 14) ÷ 2 = −11 The solution −11 is negative, and a width cannot be negative. So the width is 3 cm.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).