Functions and Graphs

Key Concepts

• A function is a rule that links each input value to exactly one output value.

  • The input is usually called x.
  • The output is usually called y.
  • We often write a function as an equation, such as:
    • $y = 2x + 3$
    • $y = x^2 - 4x + 1$

• A graph is a visual representation of the relationship between two variables.

  • The horizontal axis is the x-axis.
  • The vertical axis is the y-axis.
  • The point where the two axes meet is called the origin, $(0,0)$.

• A linear graph comes from an equation of the form:

  $$y = mx + c$$

  where:

  • $m$ is the gradient
  • $c$ is the y-intercept

• In Sec 2, you also need to know the graphs of direct proportion and inverse proportion:

  • Direct proportion: $y = kx$ — a straight line through the origin.
  • Inverse proportion: $y = \dfrac{k}{x}$ — a curve (hyperbola) that approaches but never touches the axes.

• For the direct proportion graph $y = kx$:

  • It is a straight line passing through the origin $(0, 0)$.
  • The steeper the line, the larger the value of $k$.
  • If $k > 0$, the line slopes upward to the right.

• For the inverse proportion graph $y = \dfrac{k}{x}$:

  • The graph is a curve, not a straight line.
  • As $x$ increases, $y$ decreases.
  • The curve gets closer and closer to both axes but never touches them.
  • If $k > 0$, the curve lies in the first and third quadrants.
  • If both $x$ and $y$ are positive (real-world contexts), the curve lies only in the first quadrant.

Not in Sec 2 2026: Quadratic functions ($y = ax^2 + bx + c$), parabolas, vertex, axis of symmetry, and sketching quadratic graphs are not part of the Sec 2 syllabus. These are Sec 3 E-Mathematics topics. The notes below on quadratic functions are kept for reference only.

• A quadratic function has the form:

  $$y = ax^2 + bx + c$$

  where:

  • $a$, $b$, and $c$ are numbers
  • $a \ne 0$
  • The highest power of $x$ is 2

• The graph of a quadratic function is called a parabola.

• The value of a affects the shape and direction of the parabola:

  • If $a > 0$, the parabola opens upwards.
  • If $a < 0$, the parabola opens downwards.
  • If $|a|$ is larger, the parabola is narrower.
  • If $|a|$ is smaller, the parabola is wider.

• The intercepts are where the graph crosses the axes:

  • x-intercept(s): where the graph crosses the x-axis
    • At these points, $y = 0$
  • y-intercept: where the graph crosses the y-axis
    • At this point, $x = 0$

• The vertex of a parabola is its turning point.

  • For an upward-opening parabola, the vertex is the minimum point.
  • For a downward-opening parabola, the vertex is the maximum point.

• The parabola has a line called the axis of symmetry.

  • It is a vertical line passing through the vertex.
  • The left and right sides of the parabola are mirror images of each other.

• To interpret a graph means to read information from it and explain what it shows.

  • For example:
    • whether the graph is increasing or decreasing
    • the greatest or least value
    • where it crosses the axes
    • what the shape suggests about the relationship between variables

• The gradient of a linear graph tells us how steep the line is.

  • It shows how much $y$ changes when $x$ increases by 1.

  • Formula:

    $$\text{gradient} = \frac{\text{change in } y}{\text{change in } x}$$

• The y-intercept of a linear graph is the value of $y$ when $x=0$.

  • In $y = mx + c$, the y-intercept is $c$.

Important Definitions

• Function: a rule or relationship in which each input value has exactly one output value.

• Variable: a symbol, such as $x$ or $y$, that represents a value that can change.

• Graph: a diagram that shows the relationship between two variables on axes.

• x-axis: the horizontal number line on a graph.

• y-axis: the vertical number line on a graph.

• Origin: the point $(0,0)$ where the x-axis and y-axis meet.

• Linear function: a function of the form $y = mx + c$, whose graph is a straight line.

• Gradient: the steepness of a straight line, found by $\frac{\text{change in } y}{\text{change in } x}$.

• y-intercept: the point where a graph crosses the y-axis.

• x-intercept: the point where a graph crosses the x-axis.

• Direct proportion graph: the graph of $y = kx$, a straight line through the origin.

• Inverse proportion graph: the graph of $y = k/x$, a curve that approaches the axes but never touches them.

• Quadratic function: a function of the form $y = ax^2 + bx + c$, where $a \ne 0$. (Sec 3 topic — not assessed in Sec 2 2026)

• Parabola: the U-shaped or inverted U-shaped graph of a quadratic function. (Sec 3 topic)

• Vertex: the turning point of a parabola. (Sec 3 topic)

• Axis of symmetry: the vertical line passing through the vertex that divides the parabola into two equal halves. (Sec 3 topic)

• Maximum point: the highest point on a downward-opening parabola. (Sec 3 topic)

• Minimum point: the lowest point on an upward-opening parabola. (Sec 3 topic)

• Intercept: a point where a graph crosses an axis.

• Sketch: a neat graph showing the main features clearly, without needing every point plotted exactly.

Worked Examples

Example 1: Find the gradient and y-intercept of a linear graph

Given:

Step 1: Compare with the form $y = mx + c$

So:

• $m = 3$
• $c = -5$

Step 2: State the answers

• Gradient = 3
• y-intercept = -5

Step 3: Write the y-intercept as a point

When $x=0$,

So the y-intercept is:

Answer:

• Gradient = 3
• y-intercept = $(0,-5)$

Not in Sec 2 2026: Examples 2 and 3 below involve quadratic graphs (parabolas), which are Sec 3 topics. They are kept here for reference only. Focus on Example 1 (linear graphs) for Sec 2.

Example 2: Find the intercepts of a quadratic graph

Given:

Find the y-intercept

Step 1: Let $x=0$

So the y-intercept is:

Find the x-intercepts

Step 2: Let $y=0$

Step 3: Factorise

So:

• $x=2$
• $x=3$

Therefore the x-intercepts are:

Answer:

• y-intercept = $(0,6)$
• x-intercepts = $(2,0)$ and $(3,0)$

Example 3: Find the vertex and sketch the parabola

Given:

Step 1: Identify the value of $a$

Since $a>0$, the parabola opens upwards.

Step 2: Find the intercepts

y-intercept: Let $x=0$

So y-intercept is:

x-intercepts: Let $y=0$

Factorise:

So:

• $x=1$
• $x=3$

x-intercepts are:

Step 3: Find the axis of symmetry

The axis of symmetry is midway between the two x-intercepts. Take the average of the x-intercept values:

So the axis of symmetry is $x=2$.

Step 4: Find the vertex (turning point)

The vertex lies on the axis of symmetry. Substitute $x=2$ into the equation to find the y-coordinate:

So the vertex is:

Step 5: Describe the sketch

• Upward-opening parabola
• Crosses the x-axis at $(1,0)$ and $(3,0)$
• Crosses the y-axis at $(0,3)$
• Lowest point is the vertex $(2,-1)$
• Symmetrical about the line $x=2$

Answer:

• Vertex = $(2,-1)$
• Axis of symmetry = $x=2$
• x-intercepts = $(1,0)$, $(3,0)$
• y-intercept = $(0,3)$

Common Mistakes to Avoid

• Confusing x-intercept with y-intercept

  • x-intercept: set $y=0$
  • y-intercept: set $x=0$

• Forgetting that in a quadratic function, $a \ne 0$

  • If $a=0$, the function is no longer quadratic.

• Drawing a parabola with the wrong opening direction

  • Check the sign of $a$:
    • positive → upwards
    • negative → downwards

• Plotting the vertex wrongly due to arithmetic errors when finding the average of the x-intercepts.

• Forgetting to substitute the x-value of the axis of symmetry back into the equation to find the y-coordinate of the vertex.

• Assuming every quadratic graph cuts the x-axis twice

  • Some parabolas touch the x-axis once.
  • Some do not cross the x-axis at all.

• Using unequal scales on the axes

  • This makes the graph misleading.

• Drawing the parabola with sharp corners

  • A parabola must be a smooth curve.

• Mixing up gradient and y-intercept in $y = mx + c$

  • $m$ is gradient
  • $c$ is y-intercept

• Calculating gradient incorrectly

  • Always use:

    $$\frac{\text{change in } y}{\text{change in } x}$$

  • Not the other way round.

Exam Tips

• When asked to state the gradient, write a number, not just “steep” or “positive”.

• When asked to give the y-intercept, it is best to write it as a point, for example:

  • $(0,4)$

• In sketching questions, always show the important features:

  • intercepts
  • vertex
  • axis of symmetry
  • correct opening direction

• If the equation is quadratic, check:

  • sign of $a$
  • intercepts
  • turning point

• Use mathematical keywords in your answer:

  • gradient
  • y-intercept
  • x-intercept
  • vertex
  • axis of symmetry
  • maximum
  • minimum

• For graph interpretation, mention:

  • whether the graph increases or decreases
  • the highest or lowest point
  • where the graph crosses the axes

• If the question asks for the relationship shown by a linear graph:

  • positive gradient means $y$ increases as $x$ increases
  • negative gradient means $y$ decreases as $x$ increases

• Write coordinates in the correct form:

  $$(x,y)$$

• Check your final sketch:

  • Does it match the intercepts you found?
  • Is the graph symmetrical if it is a parabola?
  • Is the shape correct?

Quick Summary

• A linear function has the form $y = mx + c$.

• In $y = mx + c$, $m$ is the gradient and $c$ is the y-intercept.

• Gradient is calculated using:

  $$\frac{\text{change in } y}{\text{change in } x}$$

• A direct proportion graph has the form $y = kx$: a straight line through the origin.

• An inverse proportion graph has the form $y = k/x$: a curve that approaches the axes but never touches them.

• For linear graph sketching, show: gradient (steepness), y-intercept, and whether the line slopes up or down.

• x-intercept is found by setting $y = 0$; y-intercept is found by setting $x = 0$.

Not in Sec 2 2026: Quadratic functions, parabolas, vertex, and axis of symmetry are Sec 3 topics. The following points are for future reference only.

• A quadratic function has the form:

  $$y = ax^2 + bx + c,\quad a \ne 0$$

• The graph of a quadratic function is a parabola.

• If $a>0$, the parabola opens upwards; if $a<0$, it opens downwards.

• The vertex is the turning point of the parabola.

• The x-coordinate of the vertex is:

  $$x=\frac{-b}{2a}$$

• The parabola is symmetrical about its axis of symmetry.

• A good sketch must show the intercepts, vertex, symmetry, and correct shape.