Graphs of functions

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What are the main types of graphs you need to know?

There are five main types of graphs (also called functions) that you must be able to recognize, draw, and work with:

1. Linear graphs (y = ax + b)

• These are straight lines
• Example: y = 2x + 3
• The graph passes through different points depending on the values of a and b
• If a is positive, the line slopes upward from left to right
• If a is negative, the line slopes downward from left to right

2. Quadratic graphs (y = ±x² + ax + b)

• These are curved lines called parabolas
• They have a U-shape (if positive) or an upside-down U-shape (if negative)
• Example: y = x² + 2x + 1 (U-shape)
• Example: y = –x² + 3x + 2 (upside-down U)
• The turning point (highest or lowest point) is called the vertex
• Quadratic graphs have a vertical line of symmetry through the vertex

3. Cubic graphs (y = ±x³ or y = ax³ + bx² + cx)

• These graphs have an S-shape or a backwards S-shape
• Example: y = x³ (passes through the origin)
• Example: y = –x³ (backwards S-shape)
• Example: y = x³ + x – 4
• They can have one or more turning points

4. Reciprocal graphs (y = a/x)

• These have two separate curved branches
• Example: y = 1/x or y = 4/x
• The branches never touch the x-axis or y-axis
• These graphs have asymptotes (lines the curve gets close to but never touches)
• For y = a/x, the asymptotes are at x = 0 (y-axis) and y = 0 (x-axis)
• If a is positive, the branches appear in the top-right and bottom-left areas
• If a is negative, the branches appear in the top-left and bottom-right areas
• You cannot substitute x = 0 into these graphs because you cannot divide by zero

5. Exponential graphs (y = ab^x + c)

• These show rapid growth or decay
• Example: y = 2^x (exponential growth)
• Example: y = (½)^x or y = 2^(–x) (exponential decay)
• Exponential growth happens when b > 1 (b is greater than 1)
  • The graph starts slowly then rises very quickly
  • Example: y = 3^x
• Exponential decay happens when 0 < b < 1 (b is between 0 and 1)
  • The graph decreases rapidly at first then flattens out
  • Example: y = (½)^x
• Both types have a horizontal asymptote (usually the x-axis, y = 0)
• The y-intercept is at (0, 1) for simple exponential graphs
• For y = ab^x + c, the asymptote is at y = c and the y-intercept is at (0, a + c)

Other important graphs

Square root graphs (y = x^(½) or y = √x)

• These start at the origin (0, 0) and curve gently upward
• They only exist for x ≥ 0 (x must be zero or positive)
• The curve increases but gets less steep as x increases

Negative power graphs (y = x^(–½) or y = 1/√x)

• Similar to reciprocal graphs but only in one quadrant
• They have asymptotes at both the x-axis and y-axis
• Example: y = 1/x² is steeper than y = 1/x and is always positive (even when x is negative)

How to construct a table of values

A table of values is a list of x-values and their corresponding y-values. You create it by substituting different x-values into the equation.

Step-by-step method:

1. Choose your x-values (these are usually given in the question)

   • For example: x = –2, –1, 0, 1, 2

2. Substitute each x-value into the equation

   • Work carefully with negative numbers
   • Always use BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction)

3. Calculate the y-value for each x-value

   • Double-check your arithmetic
   • Be especially careful with negative numbers and indices

Important tips:

• When working with negative numbers in quadratic graphs, always use brackets
  • Example: If x = –3 in y = x² + 2x, write y = (–3)² + 2(–3) = 9 – 6 = 3
• For reciprocal graphs like y = 1/x, do not include x = 0 (you cannot divide by zero)
• Use your calculator's table function to check your work if needed

Example: Create a table of values for y = x² – 3x + 1 when x goes from –1 to 3

Working:

• When x = –1: y = (–1)² – 3(–1) + 1 = 1 + 3 + 1 = 5
• When x = 0: y = 0² – 3(0) + 1 = 0 – 0 + 1 = 1
• When x = 1: y = 1² – 3(1) + 1 = 1 – 3 + 1 = –1
• When x = 2: y = 2² – 3(2) + 1 = 4 – 6 + 1 = –1
• When x = 3: y = 3² – 3(3) + 1 = 9 – 9 + 1 = 1

How to draw a graph from a table of values

Once you have your table of values, follow these steps:

1. Plot the points accurately

   • Mark each coordinate (x, y) with a small cross or dot
   • Be as precise as possible (within half a square on the grid)
   • Check the scales on both axes carefully

2. Join the points correctly

   • For curved graphs (quadratic, cubic, reciprocal, exponential): draw a smooth curve by hand through all points
   • Do NOT use a ruler for curves!
   • The curve should flow smoothly without any sharp corners or straight sections
   • For straight line graphs (linear): use a ruler to draw a straight line through the points

3. Check your graph makes sense

   • Does it have the shape you expect?
   • Does it pass through all the plotted points?
   • For quadratic graphs, is there a vertical line of symmetry?
   • If one point doesn't fit, check your calculation for that point

Understanding asymptotes

An asymptote is an invisible line that a graph approaches but never touches or crosses.

For reciprocal graphs (y = a/x):

• Vertical asymptote: x = 0 (the y-axis)
  • This is where the denominator equals zero
  • The graph cannot exist at x = 0
• Horizontal asymptote: y = 0 (the x-axis)
  • As x gets very large (positive or negative), y gets closer and closer to zero

For reciprocal graphs with a constant (y = a/x + b):

• The graph shifts up or down by b units
• Vertical asymptote: still at x = 0
• Horizontal asymptote: y = b (shifted from the x-axis)
• Example: For y = 1/x + 3, the horizontal asymptote is y = 3

For exponential graphs (y = ab^x + c):

• Horizontal asymptote: y = c
• No vertical asymptote
• Example: For y = 2^x, the asymptote is y = 0
• Example: For y = 3 × 2^x + 5, the asymptote is y = 5

Exponential growth and decay in real life

Exponential growth (y = ab^x where b > 1):

• Population increase
• Bacterial growth
• Compound interest
• The y-intercept shows the starting value
• The asymptote shows the minimum value (usually 0)
• The graph rises rapidly

Exponential decay (y = ab^x where 0 < b < 1, or y = ak^(–x) where k > 1):

• Radioactive decay
• Cooling of hot objects
• Depreciation of car values
• Medicine breaking down in the body
• The y-intercept shows the starting value
• The asymptote shows the value it settles to after a long time
• The graph falls rapidly at first, then levels off

Example: A population can be modeled as y = 400 × (½)^x + 100

• This is exponential decay because ½ is between 0 and 1
• The initial population (when x = 0) is 400 + 100 = 500
• Over time, the population settles to 100 (the asymptote)

How to recognize graphs from equations

You can identify the type of graph by looking at the equation:

Solving equations graphically

What does "solving graphically" mean?

• It means finding the x-values (solutions) by reading them from a graph
• Instead of using algebra, you use the graph to find where something happens

Finding roots (x-intercepts):

• Roots are the solutions to an equation when y = 0
• They are where the graph crosses the x-axis
• For example, the roots of y = x² – 4x + 3 are where the graph crosses the x-axis
• Read the x-values at these crossing points

Solving equations of the form: (function) = 0

• Find where the graph crosses the x-axis
• Read the x-coordinates at these points
• Example: To solve x² – 3x + 1 = 0 using the graph of y = x² – 3x + 1, find the x-intercepts

Solving equations of the form: (function) = a number

• Draw a horizontal line at that y-value
• Find where this line crosses your graph
• Read the x-coordinates at the crossing points
• Example: To solve x² – 3x + 1 = 5 using the graph of y = x² – 3x + 1
  • Draw the line y = 5
  • Find where y = 5 crosses the curve
  • Read the x-values at these points

Solving equations involving a different expression:

• Sometimes you need to solve an equation that doesn't match your graph exactly
• Rearrange the equation to get it into the form: (your graph equation) = (something simple)
• Example: If you have the graph of y = x² – 3x + 1 and need to solve x² – 3x + 5 = 0
  • Rearrange: x² – 3x + 1 = –4 (subtract 4 from both sides)
  • Now draw y = –4 on your graph
  • Find where this crosses your curve

Finding intersections of a line and a curve

When a straight line and a curve cross, the crossing points are called points of intersection.

To find intersection points:

1. Draw both graphs on the same axes

   • Plot the curve using a table of values
   • Plot the straight line (you only need two points for this)

2. Mark where they cross

   • The graphs might intersect once, twice, or not at all
   • Use a small cross or dot to mark each intersection point

3. Read the coordinates

   • Find the x-value and y-value at each intersection
   • Write them as coordinates: (x, y)

Example: Find where y = x² – 2x + 1 intersects with y = x + 1

• Plot both graphs
• They cross at two points
• Read the coordinates from the graph

Using intersections to solve simultaneous equations:

• The intersection point gives you the solution to simultaneous equations
• The x-coordinate and y-coordinate are your solutions
• Example: To solve 2x – y = 3 and 3x + y = 7 simultaneously:
  • Rearrange: y = 2x – 3 and y = –3x + 7
  • Draw both lines
  • Find the intersection point, say (2, 1)
  • The solution is x = 2, y = 1

Working with combined functions

Some functions combine different types of terms. For example:

y = x³ + x – 4 (cubic plus linear)

• This has a cubic term (x³) and a linear term (x)
• Draw it using a table of values
• It will have the general S-shape of a cubic

y = 2x + 3/x² (linear plus reciprocal)

• This combines a straight line part with a reciprocal part
• The graph will have features of both types
• Remember: x cannot be 0 (because of the 1/x² part)
• Use a table of values and plot carefully

y = ¼ × 2^x (exponential)

• This is exponential growth (because 2 > 1)
• The ¼ affects the steepness but doesn't change the basic shape
• The y-intercept is at (0, ¼)

Important things to remember

1. Always plot points accurately – within half a square on the grid
2. Use smooth curves for curved graphs – never use a ruler for curves
3. Be careful with negative numbers – always use brackets when substituting
4. Check your table of values – if a point doesn't fit, you probably made a calculation error
5. Know your asymptotes – reciprocal and exponential graphs have lines they never cross
6. Understand what you're finding – roots are x-values, intersections are (x, y) points
7. For reciprocal graphs, never include x = 0 – you cannot divide by zero
8. Read scales carefully – the axes might not go up in 1s