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CIE IGCSE Mathematics (0580) Sketching curves Study Notes

CIE IGCSE Mathematics (0580) Sketching curves Study Notes - New Syllabus

CIE IGCSE Mathematics (0580) Sketching curves Study Notes

LEARNING OBJECTIVE

  • Recognize, sketch and interpret graphs

Key Concepts: 

C2.11 Sketching curves

  • Linear
  • Quadratic Graphs

E2.11 Sketching curves

  • linear
  • quadratic
  • cubic
  • reciprocal
  • exponential.

CIE iGCSE Maths (0580)  Study Notes – All topics

Sketching Curves

Sketching Curves

(a) Linear Graphs

Linear graphs are straight lines. The general form of a linear function is:

\( y = mx + c \)

  • m is the gradient (slope) of the line.
  • c is the y-intercept (the point where the line crosses the y-axis).

To sketch a linear graph:

  • Plot the y-intercept \( (0, c) \).
  • Use the gradient to find another point (e.g., if \( m = 2 \), move up 2 units for every 1 unit across).
  • Draw a straight line through the points.

Example :

Sketch the graph of \( y = 2x – 3 \).

▶️ Answer/Explanation

Step 1: Identify gradient and intercept

  • Gradient \( m = 2 \)
  • Y-intercept \( c = -3 \), so the graph passes through \( (0, -3) \)

Step 2: Use gradient to find another point

  • From \( (0, -3) \), go up 2 units and right 1 unit → \( (1, -1) \)

Step 3: Plot the two points and draw a straight line through them

  • You may also plot a third point for accuracy: try \( x = 2 \):
  • \( y = 2(2) – 3 = 1 \) → point \( (2, 1) \)

Line passes through: (0, -3), (1, -1), and (2, 1)

   

(b) Quadratic Graphs

Quadratic graphs are parabolas. The general form is:

\( y = ax^2 + bx + c \)

  • If \( a > 0 \), the parabola opens upward.
  • If \( a < 0 \), it opens downward.
  • The graph is symmetric about a vertical line called the axis of symmetry.

Key Features to Sketch a Quadratic:

  • Roots (x-intercepts): Solve \( y = 0 \). These are the points where the graph crosses the x-axis.
  • y-intercept: Set \( x = 0 \) to find \( y = c \).
  • Axis of symmetry: Given by \( x = -\frac{b}{2a} \).
  • Vertex: The turning point of the parabola, lies on the axis of symmetry. Find its coordinates using substitution into the original equation.

To sketch:

  1. Find the roots (if factorable or by using the quadratic formula).
  2. Identify the y-intercept.
  3. Find the axis of symmetry and the vertex.
  4. Plot these key points and draw a smooth curve.

Tip:

  • A sketch shows shape and key features (roots, y-intercept, vertex), not exact points.
  • Use symmetry to plot mirrored points on either side of the vertex.

Example :

Sketch the graph of \( y = x^2 – 4x + 3 \).

▶️ Answer/Explanation

Step 1: Factorise to find roots

  • \( y = (x – 1)(x – 3) \)
  • So the graph crosses the x-axis at \( x = 1 \) and \( x = 3 \)

Step 2: Find y-intercept

  • Set \( x = 0 \): \( y = 0^2 – 4(0) + 3 = 3 \)
  • So it crosses the y-axis at \( (0, 3) \)

Step 3: Find axis of symmetry

  • Using \( x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2 \)
  • This is the line of symmetry.

Step 4: Find the vertex

  • Substitute \( x = 2 \): \( y = (2)^2 – 4(2) + 3 = 4 – 8 + 3 = -1 \)
  • Vertex is at \( (2, -1) \)

Summary of key points:

  • Roots: \( (1, 0) \), \( (3, 0) \)
  • Y-intercept: \( (0, 3) \)
  • Vertex: \( (2, -1) \)
  • Axis of symmetry: \( x = 2 \)

Sketch a symmetrical U-shaped curve through these points.

(c) Cubic Graphs

Cubic graphs have the general form:

\( y = ax^3 + bx^2 + cx + d \)

  • If \( a > 0 \), the curve rises to the right.
  • If \( a < 0 \), the curve falls to the right.
  • Cubic graphs may have 1 or 3 real roots (x-intercepts).

They often have an S-shaped curve (if 3 real roots exist), with a turning point (maximum or minimum) and a point of inflection.

Example :

Sketch the graph of \( y = x^3 – 3x \).

▶️ Answer/Explanation

Step 1: Factorise

\( y = x(x^2 – 3) = x(x – \sqrt{3})(x + \sqrt{3}) \)

Roots: \( x = 0, \pm\sqrt{3} \)

Step 2: Turning Point (approx)

The curve turns between the roots and has a point of inflection at \( x = 0 \)

Sketch: S-shaped graph passing through origin and symmetric around origin.

(d) Reciprocal Graphs

The reciprocal graph is of the form:

\( y = \frac{1}{x} \)

  • The graph has two branches (in opposite quadrants).
  • There is a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \).
  • The graph never touches the axes.

Example :

Sketch the graph of \( y = \frac{4}{x} \).

▶️ Answer/Explanation

Key points:

  • When \( x = 1 \), \( y = 4 \)
  • When \( x = -1 \), \( y = -4 \)
  • When \( x = 2 \), \( y = 2 \)
  • When \( x = -2 \), \( y = -2 \)

Asymptotes: The curve gets closer to x-axis and y-axis but never touches them.

(e) Exponential Graphs

Exponential graphs have the form:

\( y = ab^x \), where \( b > 0 \) and \( b \neq 1 \)

  • If \( b > 1 \), the graph shows exponential growth.
  • If \( 0 < b < 1 \), the graph shows exponential decay.
  • There is a horizontal asymptote at \( y = 0 \).

Example :

Sketch the graph of \( y = 2^x \).

▶️ Answer/Explanation

Key points:

  • When \( x = 0 \), \( y = 1 \)
  • When \( x = 1 \), \( y = 2 \)
  • When \( x = 2 \), \( y = 4 \)
  • When \( x = -1 \), \( y = 0.5 \)
  • As \( x \to -\infty \), \( y \to 0 \) (asymptote)

The curve starts close to the x-axis (left side) and rises steeply to the right.

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