Grade 10 & 11 Functions | Maths Lesson | 2022 | Mlungisi Nkosi

Introduction to Functions

• Functions are mathematical expressions that relate an input to an output.

• We'll cover different types of functions and practice with past exam questions.

Types of Functions

Straight Line Function

• Equation: y = mx + b

• m: Gradient (slope of the line); c: y-intercept (where the line intersects the y-axis).

• To find the gradient using two points (x1, y1) and (x2, y2):

  • Gradient (m) formula: (y2 - y1) / (x2 - x1)

• To find the y-intercept (c), substitute a known point into the equation.

Parabola Functions

1. Standard Form: y = ax² + bx + c

   • y-intercept is at (0, c).

2. Vertex Form: y = a(x - p)² + q

   • The vertex (turning point) is at (p, q). Note: p changes sign inside the bracket.

   • Example: For y = -(x + 3)² + 4, turning point = (-3, 4).

3. Factored Form: y = a(x - m)(x - n)

   • X-intercepts at (m, 0) and (n, 0). Again, change of signs observed.

Characteristics of Parabolas

• Smiling and Sad Parabolas:

  • When a > 0, the parabola opens upwards (smiling).

  • When a < 0, the parabola opens downwards (sad).

• Turning Point:

  • x-coordinate: x = -b / (2a); compute the y-coordinate by substituting back into the original equation.

• Finding X and Y Intercepts: Set y = 0 for x-intercept; set x = 0 for y-intercept.

  Key Characteristics of Parabolas:

  • Vertex: The highest or lowest point of the parabola, depending on the direction it opens.

  • Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves, given by the equation x = -b / (2a).

  • Direction: If a > 0, the parabola opens upwards (happy); if a < 0, it opens downwards (sad).

  • Focus: The width of the parabola is determined by the value of 'a'; larger values of |a| result in a narrower parabola, while smaller values lead to a wider one.

Exponential Graphs

• General form: y = a^x; crosses x-axis at (0, 1).

• Reflection: Moving to y = -a^x reflects the graph around the x-axis.

• Shift form: y = a(x - p) + q adjusts the graph horizontally and vertically.

Applications of Functions

Example Problem with Graphs

• Given functions: f(x) = -2x² + 4x + 16 and g(x) = 2x + 4

• X-intercepts determined by solving f(x) = 0.

• Turning point, vertex highest/lowest point can be calculated using -b / (2a) method.

• Graph intersections indicate common points between f and g.

Analysis and Problem Solving

• Calculate range of functions by determining maximum/minimum points.

• Use distance calculations between points on graphs.

• Understanding axis of symmetry for various functions is crucial.

Conclusion

• Review key characteristics of each function type, focusing on shape, symmetry, and intercepts.

• Practice problems related to these functions will deepen understanding.

• Future sessions will cover applications to universities and specific academic paths.