Graphs and Transformations

• What is the title of Chapter 4 in A-Level Mathematics Year 1 Pure?

  Graphs and Transformations
• What is the first section of Chapter 4?

  Cubic Graphs
• What is the second section of Chapter 4?

  Quartic Graphs
• What is the third section of Chapter 4?

  Reciprocal Graphs
• What is the fourth section of Chapter 4?

  Points of Intersection
• What is the fifth section of Chapter 4?

  Transforming Graphs
• What is the sixth section of Chapter 4?

  Combining Transformations
• What are the main types of graphs covered in Chapter 4?

  • Cubic Graphs
  • Quartic Graphs
  • Reciprocal Graphs
  • Transforming Graphs
• What is the equation for the second example of cubic graphs?

  y = x(x + 1)(x + 2)
• What is the equation for the first example of cubic graphs?

  y = (x − 2)(1 − x)(1 + x)

  
• What is the equation for the first example of quartic graphs?

  y = (x + 1)(x + 2)(x − 1)(x − 2)
• What is the equation for the second example of quartic graphs?

  y = x(x + 2)^2(3 − x)
• What is the equation for the third example of quartic graphs?

  y = (x − 1)^2(x − 3)^2
• What is the equation for the first example of reciprocal graphs?

  y = \frac{1}{x}
• What is the equation for the second example of reciprocal graphs?

  y = -\frac{3}{x}
• What is the equation for the third example of reciprocal graphs?

  y = \frac{3}{x^2}
• What are the key features of the curve C in Exam Question 1?

  • Passes through (−1, 0)
  • Touches x-axis at (2, 0)
  • Maximum at (0, 4)
• How can the equation of curve C be expressed?

  y = x^3 + ax^2 + bx + c
• What are the integer values to calculate for curve C?

  Values of a, b, and c
• What are the types of transformations discussed in Chapter 4?

  • Vertical and horizontal shifts
  • Reflections
  • Stretching and compressing
• What is the purpose of sketching transformed graphs?

  • To visualize changes in function behavior
  • To understand the effects of transformations

  
• What is the significance of points of intersection in graphs?

  • Indicates where graphs meet
  • Important for solving equations
• What is function notation used for in transformations?

  • To express transformed functions clearly
  • To simplify the representation of functions
• What is the role of reciprocal graphs in mathematics?

  • To illustrate inverse relationships
  • To analyze behavior near asymptotes
• What is the section that follows Reciprocal Graphs?

  Exponential Graphs
• What are the key aspects of sketching a transformed graph?

  • Identify transformations applied
  • Sketch original graph
  • Apply transformations step-by-step
• What is the purpose of transforming the equation of a graph?

  To analyze changes in graph shape
• What does describing from the equations involve?

  • Analyzing coefficients
  • Understanding graph behavior
• What is the process of sketching reciprocals?

  • Identify asymptotes
  • Determine intercepts
  • Sketch the curve

  
• What are the steps involved in combining transformations?

  • Identify individual transformations
  • Apply transformations in order
  • Analyze final graph shape
• What is the first example of a reciprocal graph given?

  y = $\frac{1}{x}$
• What is the third example of a reciprocal graph given?

  y = $\frac{3}{x^2}$
• What is the second example of a reciprocal graph given?

  y = $- \frac{3}{x}$
• What is the fourth example of a reciprocal graph given?

  y = $- \frac{4}{x^2}$
• What is the first example of an exponential graph given?

  y = $2^x$
• What is the second example of an exponential graph given?

  y = $3^{-x}$
• What is the process to find points of intersection of two curves?

  • Sketch the curves
  • Set equations equal: f(x) = g(x)
  • Solve for x-coordinates
• What are the equations of the curves to sketch in example 10(a)?

  y = $x(x - 3)$ and y = $x^2(1 - x)$
• What is the task in example 10(b)?

  Find coordinates of points of intersection
• What are the equations of the curves to sketch in example 11(a)?

  y = $x^2(3x - a)$ and y = $\frac{b}{x}$