A quadratic function, written as y = ax² + bx + c, graphs as a parabola — a smooth, symmetric U-shaped (or upside-down U) curve. The solutions to a quadratic equation (where y = 0) are the x-values where the parabola crosses the x-axis, called its roots. For a monic quadratic (where a = 1) with integer roots, you can solve algebraically by factorising: finding two numbers that multiply to give c and add to give b, rewriting the expression as two brackets, then setting each bracket to zero. You can also solve graphically, by plotting the curve and reading off where it crosses the x-axis, or numerically, by substituting values.
Example
To solve x² + 5x + 6 = 0: find two numbers that multiply to 6 and add to 5 — that's 2 and 3. Factorise to (x+2)(x+3) = 0, so x = -2 or x = -3. Graphing y = x² + 5x + 6 confirms this — the parabola crosses the x-axis exactly at x = -2 and x = -3.
Key terms
Parabola:
The symmetric U-shaped curve produced by graphing a quadratic function.
Root:
An x-value where a quadratic function equals zero — where its graph crosses the x-axis.
Questions
1. A quadratic function graphs as a:
Parabola
Straight line
Circle
Triangle
2. The roots of a quadratic equation are:
The x-values where it crosses the x-axis
The y-intercept only
Always negative
The steepness of the curve
3. A monic quadratic has:
a = 1
a = 0
b = 0
c = 1
4. A parabola is:
A symmetric, U-shaped curve
A straight diagonal line
A perfect circle
A random, jagged shape
5. To factorise x² + 5x + 6, you need two numbers that:
Multiply to 6 and add to 5
Add to 6 and multiply to 5
Both equal 6
Both equal 5
6. x² + 5x + 6 factorises to:
(x+2)(x+3)
(x+1)(x+6)
(x+5)(x+6)
(x−2)(x−3)
7. A quadratic equation can be solved:
Algebraically, graphically or numerically
Only algebraically, never any other way
Only by guessing randomly
Only using a ruler
8. Solve x² + 7x + 10 = 0:
x = -2 or x = -5
x = 2 or x = 5
x = -7 or x = -10
x = 7 or x = 10
9. Solve x² + 3x + 2 = 0:
x = -1 or x = -2
x = 1 or x = 2
x = -3 or x = -2
x = 3 or x = 2
10. Solve x² − 6x + 8 = 0:
x = 2 or x = 4
x = -2 or x = -4
x = 6 or x = 8
x = -6 or x = 8
11. Solve x² − x − 6 = 0:
x = 3 or x = -2
x = -3 or x = 2
x = 6 or x = -1
x = 1 or x = -6
12. A parabola that doesn't cross the x-axis at all has:
No real roots
Exactly one root
Infinite roots
A root at the origin only
13. The graph of y = x² has its lowest point (vertex) at:
(0, 0)
(1, 1)
(-1, 0)
(0, 1)
14. Factorise x² + 9x + 20:
(x+4)(x+5)
(x+2)(x+10)
(x+1)(x+20)
(x+9)(x+20)
15. Why does factorising into two brackets let you solve a quadratic equation by setting each bracket to zero?
If two factors multiply to give zero, at least one of them must itself be zero
Setting brackets to zero is just a memorised trick with no underlying reason
Factors can multiply to zero without either factor being zero
This method only works for quadratics with no real roots
16. A ball's height over time is modelled by a quadratic function. What do the roots of that function most likely represent in context?
The times when the ball is at ground level (height = 0)
The ball's maximum height
The ball's speed at any given moment
The colour of the ball
17. Why might a quadratic modelling a real-world situation (like height over time) only have one physically meaningful root, even though the equation has two mathematical solutions?
One root might correspond to a negative time or other value that doesn't make sense in the real-world context
Quadratic equations modelling real situations always have exactly one mathematical solution
Both mathematical solutions are always equally meaningful in every real-world context
Real-world quadratic models never actually have two solutions
18. Why might solving a quadratic graphically be useful for a quadratic that cannot be neatly factorised with integer roots?
Reading approximate roots from a graph works even when a clean algebraic factorisation isn't available
Graphing can only ever be used for quadratics with integer roots
Algebraic and graphical methods always give completely different, contradictory answers
Quadratics that cannot be factorised have no solutions of any kind
19. A quadratic model of profit versus price shows two roots at $2 and $18, with a maximum between them. What do the two roots most likely represent in this business context?
The prices at which the business breaks even (zero profit), before and after the most profitable price range
The exact price that generates maximum profit
Prices at which the business makes its highest possible loss
Numbers with no meaningful connection to the business context
20. Why might a company modelling profit with a quadratic function be more interested in the vertex (turning point) than in the roots?
The vertex represents the maximum (or minimum) value, such as the price that generates the greatest profit, which is often the key business question
The roots always provide more useful business information than the vertex
The vertex of a quadratic function has no practical business meaning
Roots and vertex always represent identical information for any quadratic model
21. Understanding how to graph and solve quadratic functions mainly helps you to:
Model and solve problems involving curved relationships between two variables
Assume every relationship between two variables is a straight line
Avoid ever using algebraic methods to solve equations
Treat graphing and algebraic solving as producing unrelated answers
Answer key (parent copy)
1. Parabola
2. The x-values where it crosses the x-axis
3. a = 1
4. A symmetric, U-shaped curve
5. Multiply to 6 and add to 5
6. (x+2)(x+3)
7. Algebraically, graphically or numerically
8. x = -2 or x = -5
9. x = -1 or x = -2
10. x = 2 or x = 4
11. x = 3 or x = -2
12. No real roots
13. (0, 0)
14. (x+4)(x+5)
15. If two factors multiply to give zero, at least one of them must itself be zero
16. The times when the ball is at ground level (height = 0)
17. One root might correspond to a negative time or other value that doesn't make sense in the real-world context
18. Reading approximate roots from a graph works even when a clean algebraic factorisation isn't available
19. The prices at which the business breaks even (zero profit), before and after the most profitable price range
20. The vertex represents the maximum (or minimum) value, such as the price that generates the greatest profit, which is often the key business question
21. Model and solve problems involving curved relationships between two variables