Further Graphs

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Notes

Types of Graphs

**Straight lines**: $y = mx + c (e.g., y = 3x + 2)$ . Key lines: $y = x$ and $y = -x$ .
**Horizontal lines**: $y = c (e.g., y = 4)$ .
**Vertical lines**: $x = k (e.g., x = 2)$ .
**Quadratic graphs**: $y = ax^{2} + bx + c$ . Shape is a **parabola**: u-shaped (positive a) or n-shaped (negative a).
**Reciprocal graphs**: $y = \frac{a}{x}$ . Two L-shaped branches; $x \ne 0$ . Positive a gives branches in first and third quadrants.

Key Features of Quadratic Graphs

The turning point is called the **vertex**: minimum for positive quadratics, maximum for negative quadratics.
Quadratic graphs have a vertical **line of symmetry** through the vertex: $x = k (k$ is x-coordinate of vertex).
**Roots** are x-intercepts where $y = 0$ . A quadratic can have 0, 1 (touches), or 2 roots.
Roots are symmetric about the line of symmetry.
Quadratic graphs always have one **y-intercept** $(where x = 0)$ .

Drawing Graphs from Tables

Substitute x-values into the equation to find y-values. Use brackets for negative x and follow BIDMAS.
For reciprocal graphs, **do not include $x =$ 0** (division by zero).
Plot points accurately (within half a square). Join with a **smooth freehand curve** (no ruler).
Use calculator table function: enter function, start/end x, step size. Check given y-values.
If a point doesn't fit the curve shape, check your working.

Solving Equations from Graphs

To solve $f(x) = 0$ , read the **x-intercepts** (roots) of the graph of $y = f(x)$ .
To solve $f(x) = k$ , draw horizontal line $y = k$ and read x-coordinates of intersections.
To solve $f(x) = g(x)$ , plot $y = f(x)$ and $y =$ g(x); solutions are x-coordinates of intersection points.
For equations not in the form $'graph = ...'$ , rearrange to match the given graph. $E.g., x^{2} - 4x + 3 = 1$ becomes $x^{2} - 4x - 2 = -4$ .
When solving for x, give only x-coordinates. Include y-coordinates only for simultaneous equations.

Example: Quadratic Graph from Table

Complete table for $y = x^{2} - 4x - 3: x = -2$ → $y = 9$ ? $(check: (-2)^{2} -4(-2)-3 = 4+8-3=9)$ . Given: -2→2, -1→? $(1+4-3=2)$ , 0→-3, 1→-6, 2→-7? $(4-8-3=-7)$ , 3→-6, 4→-3, 5→2.
Plot points and draw smooth u-shaped curve $for -2 \le x \le 5$ .
Solve $x^{2} - 4x - 3 = 0$ by reading x-intercepts from graph.
Line of symmetry: $x = 2 (midpoint$ of roots).

Example: Reciprocal Graph

Table for $y = \frac{15}{x}: x = -5$ → -3, -3 → -5, -2 → -7.5, -1 → -15, 1 → 15, 2 → 7.5, 3 → 5, 5 → 3.
Draw two separate branches for $x < 0$ and $x >$ 0; do not connect at $x = 0$ .
Solve $\frac{15}{x} = 6$ by drawing horizontal line $y = 6$ and reading x-coordinate of intersection $(x = 2.5)$ .
Reciprocal graphs have asymptotes at $x = 0$ and $y = 0$ .

Example: Solving with Horizontal Line

Graph $y = 1 + 5x - x^{2}$ $for -1 \le x \le 5$ . Table: x=-1→-5? $(1-5-1=-5)$ , 0→1, 1→5, 2→7, 3→7? $(1+15-9=7)$ , 4→5, 5→1.
Draw horizontal line $y = 3$ . Intersection x-coordinates solve $1+5x-x^{2} = 3$ .
Solutions: approximately $x = 0.6$ and $x = 4.4$ .
Line of symmetry: $x = 2.5$ .

Example: Solving with Another Line

Graph $y = -x^{2} + x + 5$ $for -3 \le x \le 4$ . Table: x=-3→-1, -2→3, -1→5? $(-1-1+5=3)$ , 0→5, 1→5, 2→3, 3→-1, 4→-7? $(-16+4+5=-7)$ .
Draw line $y = x$ . Intersection x-coordinates solve $-x^{2} + x + 5 = x$ → $-x^{2} + 5 = 0$ → $x = \pm \sqrt{5}$ ≈ $\pm 2.24$ .
Highest point (vertex): (0.5, 5.25). Line of symmetry: $x = 0.5$ .
Roots $of -x^{2} + x + 5 = 0$ : approximately $x = -1.8$ and $x = 2.8$ .

Quadratic Graph (Positive a)

Reciprocal Graph (Positive a)

Solving Equations from Graph

Line of Symmetry on Quadratic

Practice questions

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1.Which of the following is the equation of a straight line?
Easy
- A $y = 3x + 2$
- B $y = x^{2} + 3x + 2$
- C $y = \frac{1}{x}$
- D $y = -x^{2} + 3x + 2$
2.What is the name of the shape of a quadratic graph?
Easy
- AParabola
- BHyperbola
- CStraight line
- DCircle
3.For a quadratic graph $y = ax^{2} + bx + c$ , if a is positive, the graph is:
Easy
- Au-shaped
- Bn-shaped
- Ca straight line
- DL-shaped
4.What is the equation of the line of symmetry for a quadratic graph with vertex at (2, 5)?
Easy
- A $x = 2$
- B $y = 2$
- C $x = 5$
- D $y = 5$
5.Which of the following is a reciprocal graph?
Easy
- A $y = \frac{4}{x}$
- B $y = 4x + 1$
- C $y = x^{2} - 4$
- D $y = -x^{2} + 4$
6.What is the value of y when $x = -2$ in the equation $y = x^{2} - 3x$ ?
Easy
- A10
- B-2
- C-10
- D2
7.Which of the following is the equation of a horizontal line?
Easy
- A $y = 4$
- B $x = 4$
- C $y = x$
- D $y = -x$
8.The graph of $y = -x^{2} + 2x + 3$ is:
Easy
- An-shaped
- Bu-shaped
- Ca straight line
- DL-shaped

Summary

## Further Graphs - Quick Revision ### Types of Graphs - **Straight line**: y = mx + c (e.g. y = 3x + 2). Also y = x, y = -x, horizontal y = c, vertical x = k. - **Quadratic**: y = ax² + bx + c. Shape: parabola. Positive a → u-shape (minimum). Negative a → n-shape (maximum). Key features: vertex (turning point), vertical line of symmetry x = k, roots (x-intercepts), y-intercept. - **Reciprocal**: y = a/x (x ≠ 0). Two L-shaped branches, never cross axes. For a > 0, branches in first and third quadrants. ### Drawing Graphs from Tables - Substitute x-values into equation to get y-values. Use brackets for negatives. - Plot points accurately (within half a square). Join with smooth freehand curve (no ruler). - For quadratics, expect symmetry. For reciprocals, omit x = 0. - Calculator tip: use table function (start, end, step) to generate values. ### Solving Equations from Graphs - To solve f(x) = 0: read x-intercepts (roots). - To solve f(x) = k: draw horizontal line y = k, read x-coordinates of intersections. - To solve f(x) = g(x): plot both graphs, read x-coordinates of intersection points. - For rearranged equations, rewrite as "graph = ..." and proceed as above. ### Key Formulas - Quadratic: y = ax² + bx + c - Reciprocal: y = a/x - Line of symmetry: x = k (k is x-coordinate of vertex) ### Exam Tips - Check plotted points match given table values. - If a point doesn't fit the shape, recalculate. - When solving, give only x-values unless simultaneous equations require both coordinates.

Key terms

Quadratic graph: A graph of the form y = ax² + bx + c, producing a parabola (u-shape or n-shape).
Reciprocal graph: A graph of the form y = a/x (x ≠ 0), consisting of two L-shaped branches.
Vertex: The turning point of a quadratic graph; minimum for positive quadratics, maximum for negative quadratics.
Line of symmetry: A vertical line x = k that divides a quadratic graph into two mirror-image halves; k is the x-coordinate of the vertex.
Roots: The x-intercepts of a graph, where y = 0; solutions to the equation f(x) = 0.
y-intercept: The point where the graph crosses the y-axis (x = 0).
Table of values: A set of x and y coordinates generated by substituting x-values into an equation, used to plot the graph.
Point of intersection: A point where two graphs meet; its coordinates satisfy both equations simultaneously.

Slides

Types of Graphs

You need to recognise straight lines $(y = mx + c)$ , horizontal lines $(y = c)$ , vertical lines $(x = k)$ , quadratic graphs $(y = ax^{2} + bx + c)$ , and reciprocal graphs $(y = \frac{a}{x})$ .
Quadratic graphs are u-shaped (positive a) or n-shaped (negative a).
Reciprocal graphs have two L-shaped branches and never cross the $y-axis (x \ne 0)$ .

Which shape does a quadratic graph with a positive coefficient of $x^{2}$ have?

u-shape
n-shape
straight line
L-shaped branches

Key Features of Quadratic Graphs

The turning point is called the vertex: minimum for u-shape, maximum for n-shape.
Quadratic graphs have a vertical line of symmetry through the vertex.
They may cross the x-axis (roots) and always have one y-intercept.

Which of these are true about quadratic graphs?

They always have two x-intercepts.
They have a vertical line of symmetry.
The vertex is the maximum or minimum point.

Drawing Graphs from a Table of Values

Substitute x-values into the equation to find y-values.
Plot points accurately (within half a square) and join with a smooth curve.
For quadratics, be careful with negative numbers: use brackets and BIDMAS.
For reciprocals, do not include $x = 0 (division$ by zero).

When substituting $x = -2$ into $y = -x^{2} + 3x$ , the calculation is $y = -(____)^{2} + 3(____)$ .

Using a Calculator for Tables

Use the table function on your calculator: enter the function, start, end, and step size.
Check given y-values in the question to verify your calculations.
If a point doesn't fit the curve shape, recheck your working.

What is the first step when using a calculator to create a table of values?

Enter the step size
Type in the equation
Set the start value
Press =

Solving Equations from Graphs

To solve $f(x) = 0$ , read the x-intercepts (roots) of the graph.
To solve $f(x) = k$ , draw the horizontal line $y = k$ and read x-coordinates of intersections.
To solve $f(x) = g(x)$ , plot both graphs and find x-coordinates of intersection points.

To solve $x^{2} - 4x - 2 = 5$ using the graph of $y = x^{2} - 4x - 2$ , draw the line $y = ____ and$ read the x-coordinates of intersection.

Example: Solving from a Graph

Use the graph of $y = 10 - 8x^{2}$ to solve $10 - 8x^{2} = 0$ : read x-intercepts ≈ -1.12 and 1.12.
To solve $10 - 8x^{2} = 8$ , draw $y = 8$ and read x ≈ -0.5 and 0.5.
Only give x-coordinates as final answers unless solving simultaneous equations.

Using the graph of $y = x^{2} - 3x - 6$ , how many solutions does $x^{2} - 3x - 6 = 0$ have?

Reciprocal Graphs

Equation: $y = \frac{a}{x}$ , with $x \ne 0$ .
Graph has two L-shaped branches that never connect.
For positive a, branches are in the first and third quadrants; for negative a, reflected.

Which statements about reciprocal graphs are true?

They have a vertical asymptote at $x = 0$ .
They always cross the y-axis.
They consist of two separate curves.

Plotting Reciprocal Graphs

Do not include $x = 0$ in the table of values.
Plot points for negative and positive x separately.
Draw two smooth curves approaching the axes but never touching them.

For $y = \frac{15}{x}$ , when $x = -3, y = ____$ .

Line of Symmetry

Quadratic graphs have a vertical line of symmetry through the vertex.
Equation is $x = k$ , where k is the x-coordinate of the vertex.
For example, if vertex is (2, -3), line of symmetry is $x = 2$ .

What is the equation of the line of symmetry for a quadratic with vertex at (1, 4)?

$y = 4$
$x = 1$
$y = 1$
$x = 4$

Summary

Recognise and draw straight lines, quadratics, and reciprocals.
Use tables of values to plot points accurately.
Solve equations by reading intersections from graphs.
Identify key features: vertex, line of symmetry, roots, asymptotes.

Which graph type has a maximum point?

Positive quadratic
Negative quadratic
Reciprocal
Straight line

Flashcards

What is the general equation of a straight line?

$y = mx + c$

What is the equation of a horizontal line?

$y = c$

What is the equation of a vertical line?

$x = k$

What is the general equation of a quadratic graph?

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

What shape does a positive quadratic graph have?

A u-shape (minimum point at the bottom).

What shape does a negative quadratic graph have?

An n-shape (maximum point at the top).

What is the vertex of a quadratic graph?

The point where the graph turns; the minimum or maximum point.

What is the line of symmetry of a quadratic graph?

A vertical line through the vertex, equation $x = k$ , where k is the x-coordinate of the vertex.

What are the roots of a quadratic graph?

The x-intercepts where the graph crosses the x-axis.

What is the general equation of a reciprocal graph?

$y = \frac{a}{x}$ , where $x \ne 0$ .

What is the shape of a reciprocal graph $y = \frac{a}{x}$ for $a > 0$ ?

Two L-shaped curved branches, one in the first quadrant and one in the third quadrant.

Why $can't x = 0$ be used in a reciprocal graph?

Division by zero is undefined, so there is no value at $x = 0$ .

How do you draw a graph from a table of values?

Plot the (x, y) points accurately, then join them with a single smooth freehand curve.

What should you be careful about when substituting negative numbers into a quadratic equation?

Always put brackets around negative numbers and use BIDMAS.

How can you solve $x^{2} - 4x - 3 = 0$ using its graph?

Read the x-coordinates where the graph crosses the x-axis (the roots).

How can you solve $x^{2} - 4x - 2 = 5$ using the graph of $y = x^{2} - 4x - 2$ ?

Draw the horizontal line $y = 5$ and read the x-coordinates of the intersection points.

How can you solve $x^{2} - 4x - 2 = x + 1$ using the graph of $y = x^{2} - 4x - 2$ ?

Plot the line $y = x + 1$ on the same axes and read the x-coordinates of the intersection points.

How do you solve a different equation like $x^{2} - 4x + 3 = 1$ using the graph of $y = x^{2} - 4x - 2$ ?

Rearrange to get $'graph = ...'$ by subtracting 5 from both sides: $x^{2} - 4x - 2 = -4$ , then use the graph and line $y = -4$ .

What are the coordinates of intersection of two graphs?

The points where the two graphs meet; the x and y coordinates satisfy both equations.

How do you solve simultaneous equations graphically?

Plot both equations on the same axes, find their point(s) of intersection, and read the x and y coordinates.

What is the equation of the line of symmetry for $y = x^{2} - 3x - 6$ ?

$x = 1.5 (the x-coordinate$ of the vertex).

What is the equation of the line of symmetry for $y = 7 + 2x - x^{2}$ ?

$x = 1 (the x-coordinate$ of the vertex).

What is the highest point of a negative quadratic graph called?

The maximum point or vertex.

What is the lowest point of a positive quadratic graph called?

The minimum point or vertex.

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