Secondary 4 Elementary Mathematics Algebra Functions Quiz

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Secondary 4 Elementary Mathematics Quiz - Algebra Functions

Name: ______________________
Class: _________
Date: ___________
Score: _______ / 40

Duration: 60 minutes
Total Marks: 40
Topic: Algebra Functions (N6 Functions and Graphs)

Instructions:

• Answer all 20 questions.
• Show your working clearly where required.
• Use the answer spaces provided.
• Calculators may be used.

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Section A: Function Notation and Basic Evaluation (Questions 1–5)

1. [1 mark] Given $f(x) = 2x + 3$, find $f(4)$.

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2. [1 mark] Given $g(x) = x^2 - 1$, find $g(-2)$.

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3. [2 marks] Given $h(x) = \frac{3x - 1}{2}$, find the value of $x$ when $h(x) = 5$.

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4. [2 marks] The function $p(x) = 5 - x$. Find $p(p(2))$.

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5. [2 marks] Given $f(x) = x + 1$ and $g(x) = 2x$, find $f(g(3))$.

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Section B: Quadratic Functions in Vertex and Factored Form (Questions 6–10)

6. [1 mark] Write down the coordinates of the vertex of the graph $y = (x - 3)^2 + 2$.

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7. [2 marks] The graph of $y = -(x - 1)^2 + 4$ has a vertex at $(1, 4)$. State whether the graph opens upwards or downwards, and find the $y$-intercept.

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8. [2 marks] For the quadratic function $y = (x - 2)(x - 5)$, find the $x$-intercepts and the equation of the axis of symmetry.

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9. [2 marks] Sketch the graph of $y = -(x + 1)(x - 3)$ on the grid below. Mark the $x$-intercepts and vertex.

Image pending generation: graph for Q9.

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10. [2 marks] Given $y = (x - p)^2 + q$ has vertex $(2, -3)$, write the equation and find the $y$-intercept.

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Section C: Power and Exponential Functions (Questions 11–15)

11. [1 mark] State the shape of the graph of $y = x^2$ for $x > 0$.

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12. [2 marks] Given $y = ax^n$ where $a = 2$ and $n = -1$, write the equation and state one feature of its graph.

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13. [2 marks] The exponential function $y = 2 \cdot 3^x$ passes through $(0, k)$. Find $k$ and state the value of $y$ when $x = 2$.

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14. [2 marks] For $y = x^3$, find $y$ when $x = -2$ and sketch the general shape.

Image pending generation: graph for Q14.

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15. [2 marks] Explain why the graph of $y = x^{-2}$ is always positive for $x \ne 0$.

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Section D: Gradient of a Curve and Mixed Application (Questions 16–20)

16. [2 marks] The table shows values for $y = x^2$:

$x$	1	2	3
$y$	1	4	9

Use the values at $x = 2$ and $x = 3$ to estimate the gradient of the curve at $x = 2.5$.

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17. [2 marks] On the grid below, draw a tangent to the curve $y = x^2$ at $x = 1$ and use it to estimate the gradient.

Image pending generation: graph for Q17.

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18. [2 marks] Given $f(x) = 3x - 2$ and $g(x) = x^2$, find $g(f(1))$.

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19. [2 marks] A function is given by $y = 100 \cdot 2^x$, where $x$ is time in hours. Find $y$ when $x = 3$ and explain what this represents.

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20. [3 marks] The quadratic $y = (x - a)(x - b)$ has $x$-intercepts at $1$ and $4$.
(a) Write the equation. [1]
(b) Find the axis of symmetry. [1]
(c) Find the vertex coordinates. [1]

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Secondary 4 Elementary Mathematics Quiz - Algebra Functions (Answer Key)

Topic: Algebra Functions (N6 Functions and Graphs)
Total Marks: 40

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Section A: Function Notation and Basic Evaluation

1. [1 mark]
$f(x) = 2x + 3$
$f(4) = 2(4) + 3 = 8 + 3 = 11$
Answer: $11$
Teaching note: Substitute $x = 4$ into the expression. Function notation $f(4)$ means "value of the function when $x = 4$".

2. [1 mark]
$g(x) = x^2 - 1$
$g(-2) = (-2)^2 - 1 = 4 - 1 = 3$
Answer: $3$
Teaching note: Squaring a negative gives positive: $(-2)^2 = 4$.

3. [2 marks]
$h(x) = \frac{3x - 1}{2} = 5$
$3x - 1 = 10$ (multiply both sides by 2)
$3x = 11$
$x = \frac{11}{3}$
Answer: $x = \frac{11}{3}$
Marking: 1 mark for correct equation setup, 1 mark for correct $x$.

4. [2 marks]
$p(x) = 5 - x$
$p(2) = 5 - 2 = 3$
$p(p(2)) = p(3) = 5 - 3 = 2$
Answer: $2$
Teaching note: Evaluate inside-out: first $p(2)$, then apply $p$ again.

5. [2 marks]
$g(3) = 2(3) = 6$
$f(g(3)) = f(6) = 6 + 1 = 7$
Answer: $7$
Marking: 1 mark for $g(3)$, 1 mark for final $f(6)$.

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Section B: Quadratic Functions in Vertex and Factored Form

6. [1 mark]
$y = (x - 3)^2 + 2$ is vertex form $y = (x - p)^2 + q$ with $p = 3, q = 2$.
Answer: $(3, 2)$

7. [2 marks]
$y = -(x - 1)^2 + 4$: negative sign means opens downwards.
$y$-intercept: set $x = 0$: $y = -(0 - 1)^2 + 4 = -1 + 4 = 3$.
Answer: Downwards; $y$-intercept $= 3$
Marking: 1 mark direction, 1 mark intercept.

8. [2 marks]
$y = (x - 2)(x - 5)$: $x$-intercepts at $x = 2$ and $x = 5$.
Axis of symmetry: $x = \frac{2 + 5}{2} = 3.5$.
Answer: $x$-intercepts $2, 5$; axis $x = 3.5$

9. [2 marks]
$y = -(x + 1)(x - 3)$: $x$-intercepts $-1, 3$; vertex at $x = 1$, $y = -(2)(-2) = 4$.
Sketch: downward parabola through $(-1,0), (3,0), (1,4)$.
Marking: 1 mark intercepts correct, 1 mark vertex and shape.

10. [2 marks]
Vertex $(2, -3)$ → $y = (x - 2)^2 - 3$.
$y$-intercept: $x = 0$: $y = (-2)^2 - 3 = 4 - 3 = 1$.
Answer: $y = (x - 2)^2 - 3$; $y$-intercept $1$

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Section C: Power and Exponential Functions

11. [1 mark]
$y = x^2$ for $x > 0$ is a curve increasing from origin, symmetric, U-shaped.
Answer: Increasing curve (parabola branch) in first quadrant.

12. [2 marks]
$y = 2x^{-1} = \frac{2}{x}$. Feature: graph has two branches, asymptotic to axes, never touches $x=0$.
Answer: $y = \frac{2}{x}$; one feature e.g. "vertical asymptote at $x=0$".

13. [2 marks]
$x = 0$: $y = 2 \cdot 3^0 = 2(1) = 2$ → $k = 2$.
$x = 2$: $y = 2 \cdot 3^2 = 2 \cdot 9 = 18$.
Answer: $k = 2$; $y = 18$

14. [2 marks]
$x = -2$: $y = (-2)^3 = -8$. Shape: S-curve through origin, negative for negative $x$.
Answer: $-8$; sketch as cubic through $(-2,-8),(0,0),(2,8)$.

15. [2 marks]
$y = x^{-2} = \frac{1}{x^2}$. Since $x^2 > 0$ for all $x \ne 0$, $\frac{1}{x^2} > 0$.
Answer: Always positive because square of non-zero real is positive, reciprocal positive.

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Section D: Gradient of a Curve and Mixed Application

16. [2 marks]
Gradient estimate $= \frac{9 - 4}{3 - 2} = \frac{5}{1} = 5$.
Answer: $5$
Teaching note: Uses chord between two points near $x=2.5$ as approximation.

17. [2 marks]
At $x=1$, $y=1$. Tangent slope ≈ derivative $2x$ at $x=1$ is $2$. Drawing tangent gives ~2.
Answer: Gradient ≈ $2$
Marking: 1 mark tangent drawn, 1 mark estimate.

18. [2 marks]
$f(1) = 3(1) - 2 = 1$
$g(f(1)) = g(1) = 1^2 = 1$
Answer: $1$

19. [2 marks]
$x = 3$: $y = 100 \cdot 2^3 = 100 \cdot 8 = 800$.
Represents quantity after 3 hours doubling each hour from 100.
Answer: $800$; initial 100 doubles every hour.

20. [3 marks]
(a) $y = (x - 1)(x - 4)$ [1]
(b) Axis: $x = \frac{1+4}{2} = 2.5$ [1]
(c) Vertex: $x=2.5$, $y = (1.5)(-1.5) = -2.25$ → $(2.5, -2.25)$ [1]
Answer: (a) $y=(x-1)(x-4)$ (b) $x=2.5$ (c) $(2.5, -2.25)$