Further Graphs And Tangents

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Notes

Types of Graphs

**Linear**: $y = mx + c$ or $ax + by = c (straight line)$ .
**Quadratic**: $y = ax^{2} + bx + c (parabola, symmetric)$ .
**Cubic**: $y = ax^{3} + bx^{2} + cx (S-shaped curve)$ .
**Reciprocal**: $y = \frac{a}{x} + b (two$ branches, asymptotes at $x=0$ and $y=b)$ .
**Exponential**: $y =$ a·kˣ + b (growth if $k>1$ , decay if 0<k<1; asymptote at $y=b)$ .
**Trigonometric**: sine, cosine, tangent (periodic).

Drawing Graphs from Tables

Substitute x-values into equation to get y-values; avoid $x=0$ for reciprocal graphs.
Plot points accurately (within half a square) and join with a smooth freehand curve.
Use calculator table function: enter function, start/end x, step size.
Check for symmetry (e.g., quadratics have vertical line of symmetry).
Be careful with negative numbers: use brackets and BIDMAS.

Solving Equations from Graphs

Solutions to $f(x)=0$ are x-intercepts (roots) of the graph.
To solve $f(x)=k$ , draw horizontal line $y=k$ and read intersection x-coordinates.
To solve $f(x)=g(x)$ , plot both graphs; intersection x-values are solutions.
Rearrange given equation to match the graph's equation plus a line.
Only give x-coordinates unless solving simultaneous equations (include y).

Finding Gradients of Tangents

Gradient of curve at a point = gradient of tangent at that point.
Draw tangent by eye using a ruler; extend line for accuracy.
Calculate gradient $= rise/run =$ Δy/Δx using two far-apart points on tangent.
Gradient represents rate of change (e.g., speed from distance-time graph).
Estimation is approximate; exact gradient requires differentiation.

Key Features of Reciprocal Graphs

$y = \frac{a}{x}$ has vertical asymptote at $x=0$ and horizontal asymptote at $y=0$ .
$y = \frac{a}{x} + b$ shifts horizontal asymptote to y=b; vertical asymptote remains $x=0$ .
$y = \frac{1}{x}^{2}$ is always positive and steeper than $y = \frac{1}{x}$ .
No y-intercept or roots for $y = \frac{a}{x}$ .

Key Features of Exponential Graphs

$y =$ kˣ $(k>1)$ shows exponential growth; y-intercept at (0,1); asymptote $y=0$ .
$y =$ kˣ $(0<k<1)$ shows exponential decay; same intercept and asymptote.
$y =$ a·kˣ + b has y-intercept at (0, a+b) and horizontal asymptote at $y=b$ .
Negative powers (e.g., 2⁻ˣ) also represent decay.

Types of Graphs Overview

Drawing a Tangent to Estimate Gradient

Solving Equations Graphically

Reciprocal Graph Asymptotes

Practice questions

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1.Which of the following is the equation of a reciprocal graph?
Easy
- A $y = x^{2}$
- B $y = \frac{1}{x}$
- C $y = 2^{x}$
- D $y = x^{3}$
2.What is the y-intercept of the graph $y = 2^{x}$ ?
Easy
- A(0, 0)
- B(0, 1)
- C(0, 2)
- D(1, 2)
3.Which type of graph has a vertical asymptote at $x = 0$ ?
Easy
- ALinear
- BQuadratic
- CReciprocal
- DExponential
4.The graph of $y = x^{2}$ is called a
Easy
- Astraight line
- Bparabola
- Ccubic curve
- Dhyperbola
5.Which of the following graphs represents exponential decay?
Medium
- A $y = 2^{x}$
- B $y = (\frac{1}{2})^{x}$
- C $y = x^{2}$
- D $y = \frac{1}{x}$
6.The graph of $y = x^{3} + 2x^{2} - x - 2$ is sketched. How many times does it cross the x-axis?
Medium
- A0
- B1
- C2
- D3
7.What is the horizontal asymptote of $y = \frac{3}{x} - 2$ ?
Medium
- A $y = 0$
- B $y = -2$
- C $x = 0$
- D $y = 3$
8.The table shows values for $y = 2x^{2} - 3x + 1$ . Which y-value is missing? x: 0, 1, 2 y: 1, 0, ?
Medium
- A1
- B3
- C5
- D7

Summary

## Further Graphs & Tangents – Quick Revision ### Types of Graphs - **Linear**: $y = mx + c$ or $ax + by = c$ - **Quadratic**: $y = ax^2 + bx + c$ (U-shaped, symmetrical) - **Cubic**: $y = ax^3 + bx^2 + cx + d$ (S-shaped) - **Reciprocal**: $y = \frac{a}{x}$ (two branches, asymptotes at $x=0$ and $y=0$) - **Exponential**: $y = a k^x + b$ (growth if $k>1$, decay if $0<k<1$; asymptote at $y=b$) - **Trigonometric**: sine, cosine, tangent (see Trigonometry notes) ### Drawing Graphs from Tables - Substitute $x$ values into equation to get $y$. - Plot points accurately (within half a square). - Join with a smooth freehand curve (no ruler for curves!). - Watch out: - For $y = \frac{1}{x}$ or $y = \frac{1}{x^2}$, $x=0$ gives no value (vertical asymptote). - Use brackets for negative numbers, e.g. $x=-3$ in $y = -x^2 + 8x$: $y = -(-3)^2 + 8(-3) = -9 -24 = -33$. ### Solving Equations from Graphs - **Intersection points**: Solve two equations simultaneously by plotting both graphs; the $x$ (and $y$) coordinates of intersection are the solutions. - **Single equation**: Given graph of $y=f(x)$, to solve $f(x)=c$, draw horizontal line $y=c$ and read $x$ at intersections. - **Rearranging**: If given a different equation, rearrange to $f(x) = mx + c$ and draw that line. - Only give $x$-coordinates unless asked for both. ### Finding Gradients of Tangents - Gradient of curve at a point = gradient of tangent at that point. - **To estimate**: 1. Draw tangent using a ruler (make it long). 2. Choose two points far apart on the tangent. 3. Gradient = $\frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}$. - The gradient represents rate of change (e.g. speed from distance-time, acceleration from speed-time). ### Key Tips - Reciprocal graphs: $y = \frac{a}{x} + b$ shifts asymptote to $y=b$. - Exponential: $y = a k^x + b$ has asymptote $y=b$, $y$-intercept $(0, a+b)$. - For cubic graphs, sketch by finding roots and $y$-intercept. - When drawing tangents, use a ruler and pick easy coordinates for gradient calculation.

Key terms

Asymptote: A line that a curve approaches but never touches. Reciprocal graphs have vertical asymptote at x=0 and horizontal asymptote at y=0 (or y=b if shifted).
Tangent: A straight line that touches a curve at a point without crossing it. Its gradient equals the gradient of the curve at that point.
Gradient: The steepness of a line, calculated as rise/run (change in y divided by change in x). For a curve, it is estimated using the tangent.
Root (x-intercept): A point where the graph crosses the x-axis (y=0). Solutions to f(x)=0 are the x-coordinates of roots.
Point of intersection: A point where two graphs meet. Its coordinates satisfy both equations simultaneously.
Reciprocal graph: Graph of y = a/x (or y = a/x + b). Has two branches, no y-intercept, asymptotes at x=0 and y=0 (or y=b).
Exponential graph: Graph of y = a k^x + b. Shows growth (k>1) or decay (0<k<1). Has horizontal asymptote at y=b and y-intercept at (0, a+b).

Slides

Types of Graphs

You need to recognise, sketch, and interpret linear, quadratic, cubic, reciprocal, exponential, and trigonometric graphs.
Linear: $y = mx + c$ or $ax + by = c$
Quadratic: $y = ax^{2} + bx + c$
Cubic: $y = ax^{3} + bx^{2} + cx$
Reciprocal: $y = \frac{a}{x} + b$
Exponential: $y =$ akˣ + b

Which type of graph does $y = \frac{3}{x}$ represent?

Linear
Quadratic
Reciprocal
Exponential

Drawing Graphs from Tables

Substitute x-values into the equation to get y-values.
Plot points accurately (within half a square) and join with a smooth curve.
Be careful with negative numbers: use brackets and BIDMAS.
For reciprocal graphs, $x=0$ gives no value (vertical asymptote).

When drawing a reciprocal graph like $y = \frac{1}{x}$ , there is a vertical asymptote at $x = ____$ .

Solving Equations from Graphs

The solutions to an equation are the x-coordinates where the graph meets the line $y =$ constant.
For equations like $f(x) = g(x)$ , plot both graphs and find intersection points.
Rearrange the equation so that one side is the given graph.

To solve $x^{2} - 4x - 2 = 0$ using the graph of $y = x^{2} - 4x - 2$ , what do you look for?

The y-intercept
The x-intercepts
The turning point
The line of symmetry

Finding Gradients of Tangents

The gradient of a curve at a point equals the gradient of the tangent at that point.
Draw a tangent (a straight line that just touches the curve).
Calculate gradient $= rise/run$ using two points far apart on the tangent.
This gives an estimate of the rate of change.

Which of these are true about drawing a tangent?

The tangent should cross the curve.
Use a ruler to draw the tangent.
Pick two points close together to find the gradient.
The gradient of the tangent equals the gradient of the curve at that point.

Exponential Graphs

$y =$ kˣ shows exponential growth if $k > 1$ , decay if $0 < k < 1$ .
Horizontal asymptote at $y = 0 (or y = b$ if shifted).
y-intercept at (0, 1) for $y =$ kˣ, or (0, a+b) for $y =$ akˣ + b.

What is the horizontal asymptote of $y =$ 2ˣ + 3?

$y = 0$
$y = 2$
$y = 3$
$y = 5$

Using Graphs to Solve Simultaneous Equations

Plot both equations on the same axes.
The intersection point gives the x and y solutions.
Rearrange into $y = mx + c$ if needed.

To solve $y = x^{2} - 2x$ and $y = x + 1$ graphically, find $the ____ of$ intersection.

Sketching Graphs

Label axes and key points: intercepts, turning points, asymptotes.
For cubic graphs, note the shape: positive coefficient goes up then down then up.
For reciprocal graphs, branches approach asymptotes.

Which graph has a vertical asymptote at $x = 0$ ?

$y = x^{2}$
$y = \frac{1}{x}$
$y =$ 2ˣ
$y = x^{3}$

Tangents and Gradients

The gradient of a tangent is an estimate of the rate of change.
In distance-time graphs, gradient = speed.
In speed-time graphs, gradient = acceleration.

Which statements about gradients are true?

The gradient of a distance-time graph is speed.
The gradient of a speed-time graph is distance.
A negative gradient means the graph is decreasing.
The gradient of a curve is constant.

Worked Example: Drawing a Graph

Complete a table of values for $y = 10 - 8x^{2}$ from $x = -1.5$ to 1.5.
Plot points: (-1.5, -8), (-1, 2), (-0.5, 8), (0, 10), (0.5, 8), (1, 2), (1.5, -8).
Join with a smooth curve. The line of symmetry is $x = 0$ .

The line of symmetry of $y = 10 - 8x^{2}$ is $x = ____$ .

Worked Example: Solving Equations

Use the graph of $y = x^{3} + x^{2} - 3x - 1$ to solve $x^{3} + x^{2} - 4x = 0$ .
Rearrange: $x^{3} + x^{2} - 3x - 1 = x - 1$ .
Plot $y = x - 1$ on the same axes.
Intersection x-coordinates: approx -2.6, 0, 1.6.

How many solutions does $x^{3} + x^{2} - 4x = 0$ have?

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