1. 1 Everything Maths www.everythingmaths.co.za 5. Functions Grade 10

2. 2 Everything Maths www.everythingmaths.co.za Functions ● A function is a mathematical relationship between two variables, where every input variable has one output variable. ● The x-variable is known as the input or independent variable, because its value can be chosen freely. ● The calculated y-variable is known as the output or dependent variable, because its value depends on the chosen input value. ● An asymptote is a straight line, which the graph of a function will approach, but never touch. ● A graph is said to be continuous if there are no breaks in the graph.

3. 3 Everything Maths www.everythingmaths.co.za Different notation ● Set notation : ● Interval notation: ● Function notation:

4. 4 Everything Maths www.everythingmaths.co.za Domain and range ● Domain: the set of all independent x-values for which there is one dependent y-value according to that function. ● Range: the set of all dependent y-values which can be obtained using an independent x-value.

5. 5 Everything Maths www.everythingmaths.co.za Linear functions y = f(x) = x ● Basic equation:y = x ● Intercept: (0; 0) ● Domain: x ∈ R ● Range: f(x) ∈ R

6. 6 Everything Maths www.everythingmaths.co.za Linear functions y = f(x) = mx + c ● Equations of the form:y = mx + c ● Intercepts: Let x = 0,(0; c) Let y = 0,(x; 0) ● Domain: x ∈ R ● Range: f(x) ∈ R

7. 7 Everything Maths www.everythingmaths.co.za Linear functions y = mx + c ● The gradient of a line is determined by the ratio of vertical change to horizontal change: ● The effects of m and c:

8. 8 Everything Maths www.everythingmaths.co.za Linear functions y = f(x) = mx + c ● Dual intercept method: 1. Determine y-intercept; let x = 0 and solve. 2. Determine x-intercept; let y = 0 and solve. 3. Use the two intercepts to sketch the graph. ● Gradient and y-intercept method: 1. Determine y-intercept; let x = 0 and solve. 2. Determine gradient, equation must be in standard form y = mx + c 3. Use the y-intercept and gradient to determine second point on the line.

9. 9 Everything Maths www.everythingmaths.co.za Quadratic functions y = f(x) = x 2 ● Basic equation:y = x 2 ● Intercepts: (0; 0) ● Domain: x ∈ R ● Range: {y : y ∈ R, y ≥ 0} ● Axis of symmetry: x = 0 ● Turning point: (0; 0)

10. 10 Everything Maths www.everythingmaths.co.za Quadratic functions y = f(x) = ax 2 + q ● Equations of the form: y = ax 2 + q ● Intercepts: Let x = 0,(0; q) Let y = 0,0 = ax 2 + q and solve for x ● Domain: x ∈ R ● Range: if a > 0, f(x) > q if a < 0, f(x) < q ● Axis of symmetry: x = 0 ● Turning point: (0; q)

11. 11 Everything Maths www.everythingmaths.co.za Quadratic functions y = ax 2 + q ● The effects of a and q:

12. 12 Everything Maths www.everythingmaths.co.za Hyperbolic functions ● Basic equation: ● Intercepts: none ● Domain: {x : x ∈ R, x ≠ 0} ● Range: {y : y ∈ R, y ≠ 0} ● Axis of symmetry: y = x and y = -x ● Asymptotes: horizontal y = 0 vertical x = 0

13. 13 Everything Maths www.everythingmaths.co.za Hyperbolic functions ● Equations of the form: ● Intercepts: Let x = 0,no y-intercept Let y = 0,and solve for x ● Domain: {x : x ∈ R, x ≠ 0} ● Range: {y : y ∈ R, y ≠ q} ● Axis of symmetry: y = x + q and y = -x + q ● Asymptotes: horizontal y = q vertical x = 0

14. 14 Everything Maths www.everythingmaths.co.za Hyperbolic functions ● The effects of a and q:

15. 15 Everything Maths www.everythingmaths.co.za Exponential functions y = f(x) = b x ● Basic equation: y = b x, b > 0 and b≠ 1. ● Intercepts: Let x = 0,(0; 1) Let y = 0,no x-intercept ● Domain: x ∈ R ● Range: {y : y ∈ R, y > 0} ● Asymptote: y = 0

16. 16 Everything Maths www.everythingmaths.co.za Exponential functions y = f(x) = ab x + q ● Equation of the form:y = ab x + q ● Intercepts: Let x = 0,(0; a + q) Let y = 0,and solve for x ● Domain: x ∈ R ● Range: if a > 0, f(x) > q if a < 0, f(x) < q ● Axis of symmetry: y = x + q and y = -x + q ● Asymptotes: horizontal y = q

17. 17 Everything Maths www.everythingmaths.co.za Exponential functions y = f(x) = ab x + q ● The effects of a and q:

18. 18 Everything Maths www.everythingmaths.co.za Sine functions y = f(x) = sin θ ● Basic equation:y = sin θ ● Domain: [ 0 0 ; 360 0 ] ● Range: [−1; 1] ● x-intercepts: (0 0 ; 0), (180 0 ; 0), (360 0 ; 0) ● y-intercept: (0 0 ; 0) ● Maximum turning point: (90 0 ; 1) ● Minimum turning point: (270 0 ; −1) ● Period: 360 0 ● Amplitude: 1

19. 19 Everything Maths www.everythingmaths.co.za Sine functions y = f(x) = a sin θ + q ● Equation of the form:y = a sin θ + q ● Domain: [ 0 0 ; 360 0 ] ● Range: if a > 0, f (θ) ∈ [−a + q, a + q] if a < 0, f (θ) ∈ [a + q, −a + q] ● Period: 360 0

20. 20 Everything Maths www.everythingmaths.co.za Sine functions y = f(x) = a sin θ + q ● The effects of a and q

21. 21 Everything Maths www.everythingmaths.co.za Cosine functions y = f(x) = cos θ ● Basic equation:y = cos θ ● Range: [−1; 1] ● x-intercepts: (90◦ ; 0), (270◦ ; 0) ● y-intercept: (0◦ ; 1) ● Maximum turning points: (0◦ ; 1), (360◦ ; 1) ● Minimum turning point: (180◦ ; −1) ● Period: 360 0 ● Amplitude: 1

22. 22 Everything Maths www.everythingmaths.co.za Cosine functions y = f(x) = a cos θ + q ● Equation of the form:y = a cos θ + q ● Domain: [ 0 0 ; 360 0 ] ● Range: if a > 0, f (θ) ∈ [−a + q, a + q] if a < 0, f (θ) ∈ [a + q, −a + q] ● Period: 360 0

23. 23 Everything Maths www.everythingmaths.co.za Cosine functions y = f(x) = a cos θ + q ● The effects of a and q:

24. 24 Everything Maths www.everythingmaths.co.za Comparison of graphs of y = sin θ and y = cos θ

25. 25 Everything Maths www.everythingmaths.co.za Tangent functions y = f(x) = tan θ ● Basic equation: y = tan θ ● Asymptotes: the lines θ = 90◦ and θ = 270◦ ● Period: 180◦ ● Domain: {θ : 0◦ ≤ θ ≤ 360◦, θ ≠ 90◦ ; 270◦ } ● Range: f (θ) ∈ R ● x-intercepts: (0◦ ; 0), (180◦ ; 0), (360◦ ; 0) ● y-intercept: (0◦ ; 0)

26. 26 Everything Maths www.everythingmaths.co.za Tangent functions y = f(x) = a tan θ + q ● Equation of the form: y = a tan θ + q ● Asymptotes: the lines θ = 90◦ and θ = 270◦ ● Period: 180◦ ● Domain: {θ : 0◦ ≤ θ ≤ 360◦, θ ≠ 90◦ ; 270◦ } ● Range: f(θ) ∈ R

27. 27 Everything Maths www.everythingmaths.co.za Tangent functions y = f(x) = a tan θ + q ● The effects of a and q:

28. 28 Everything Maths www.everythingmaths.co.za For more practice or to ask an expert for help on this section see: www.everythingmaths.co.za Shortcode: EMAAM