# Quadratic functions, equations and inequalities · Grade 9

> Explanation, worked examples and free practice following the Israeli curriculum.

In Quadratic functions, equations and inequalities (Grade 9) you practise 10 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 · Solving with the quadratic formula · Rearranging, then solving · Problems that lead to a quadratic equation · Solving an inequality from the graph · Solving a quadratic inequality.

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

- Example: The function is f(x) = −3x² . Work out f(5) .
- Answer: −75
- Solution: f(5) = (−3) × 5² = (−3) × 25 = −75

### The turning point: minimum or maximum

- Example: What are the coordinates of the turning point of the parabola y = 3x² − 7 ?
- Answer: (0, −7)
- Solution: 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

- Example: (A coordinate grid with a parabola) The picture shows the graph of a quadratic function. What are the coordinates of the vertex of the parabola?
- Answer: (−4, −2)
- Solution: (−4, −2)

### Factorised form and where the graph crosses the axis

- Example: The function is f(x) = (x + 7)(x − 9) . Where does its graph cross the horizontal axis? Write the values of x.
- Answer: x = −7 or x = 9
- Solution: x + 7 = 0 x = −7 x − 9 = 0 x = 9

### Matching a rule to a graph

- Example: (A coordinate grid with a parabola) Which rule fits the graph?
- Options: y = −2x² − 5 · y = 2x² + 5 · y = −2x² + 5 · y = 2x² − 5
- Answer: y = −2x² + 5
- Solution: The parabola opens downwards, and its turning point is (0, 5) . The rule that fits: y = −2x² + 5

### Solving with the quadratic formula

- Example: Solve: x² − 2x − 63 = 0
- Answer: x = 9 or x = −7
- Solution: The expression under the square root in the quadratic formula: (−2)² − 4 × 1 × (−63) = 256 √256 = 16 x = (2 + 16) ÷ 2 = 9 x = (2 − 16) ÷ 2 = −7

### Rearranging, then solving

- Example: Solve: x² = 2x + 63
- Answer: x = 9 or x = −7
- Solution: Move every term to one side: x² − 2x − 63 = 0 The expression under the square root in the quadratic formula: (−2)² − 4 × 1 × (−63) = 256 √256 = 16 x = (2 + 16) ÷ 2 = 9 x = (2 − 16) ÷ 2 = −7

### Problems that lead to a quadratic equation

- Example: A rectangle is 8 cm longer than it is wide, and its area is 33 cm². How wide is it?
- Answer: 3 cm
- Solution: 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.

### Solving an inequality from the graph

- Example: (A coordinate grid with a parabola) The picture shows the graph of y = f(x) . Solve the inequality f(x) > 0
- Options: x < −5 or x > −1 · −5 < x < −1 · x ≤ −5 or x ≥ −1 · −5 ≤ x ≤ −1
- Answer: x < −5 or x > −1
- Solution: The graph crosses the horizontal axis where x = −5, x = −1 . The parabola opens upwards, so it is above the horizontal axis on both sides, outside the crossing points. The solution: x < −5 or x > −1

### Solving a quadratic inequality

- Example: Solve: (x + 6)(x − 8) > 0
- Options: x < −6 or x > 8 · −6 < x < 8 · Every number · No solution
- Answer: x < −6 or x > 8
- Solution: Each bracket is 0 at one point: x = −6, x = 8 The parabola opens upwards, so it is above the horizontal axis on both sides, outside the crossing points. The solution: x < −6 or x > 8

## Not on the site yet

- Sketching the graph of a quadratic function by hand
- Completing the square
- Choosing the quickest method
- Equations with a parameter
- Inequalities of higher degree
- Rational inequalities

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education's grade 9 maths curriculum (2026–27).

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