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# Algebraic Graphs

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## Algebraic Graphs

Obviously, to plot co-ordinates on a graph you need x-values and y-values so, unlike when you are solving equations, we will be using y = .... to give us our co-ordinates. This means that they are called **functions **rather than equations.

**Key points**

**1.** Always make a **table** of values before plotting a graph.

**2.** Always do everything in sharp pencil - sharp for accuracy (very important to examiners!) and pencil because if you mess it up on an exam paper you must be able to rub the whole thing out as there won't be another space for you to try again!

**3.** Make sure your axes are sensibly labelled with appropriate scales.

**4.** If the points are in a straight line, you must draw a straight line **through** them to fill the whole of your graph. Don't just start at the first point plotted and end at the last.

**5.** If the points are not in a straight line, draw a **smooth curve** through them. If you're not very good at this you must practise to get your technique sorted out!

**6.** Always **label** your graph fully - the axes and the line or curve (with it's equation).

**7.** If there is a rogue point that doesn't look like it follows the line or curve then you've probably worked it out wrong and need to check it!

Linear functions can be written in the form

**y = mx + c**

where y and x are variables, m and c are constants (numbers).

If you write them like this then **m** is the **gradient** and **c** is the **y-intercept** (point where it crosses the y-axis). The graphs of Linear Functions are **straight lines.**

First make a table for x and y. You need to pick 4 or 5 values of x depending on your scale and use the function to find the corresponding values of y. Usually values around the origin will do.

Here's an example with the function **y = 2x - 1.** We will use the x values -3, -2, 0, 2 and 4.

**When** x = -3, y = -6 - 1 = -7

**When** x = -2, y = -4 - 1 = -5

**When** x = 0, y = -1

**When** x = 2, y = 4 - 1 = 3

**When** x = 4, y = 8 - 1 = 7

x |
-3 -2 0 2 4 |

y |
-7 -5 -1 3 7 |

**Now simply decide on your axes, plot the points and draw a straight line right through them:**

**We know that the equation of a straight line must be able to be written in the form:**

**y = mx + c**

So, we simply need to find **m** and **c.**

**To find m:**

**1. m** is the gradient or slope. Pick any two points on your line (but try to make them whole numbered co-ordinates so they're easy to use). **Then:**

Make sure you specify whether the change is **negative** or **positive.**

**2.** Simplify your fraction. Often it will be a whole number anyway!

**To find c:**

**1. c** is simply the place where the line crosses the y-axis.

**2.** Put m and c into y = mx + c and you're finished.

The graph below is of y = 2x + 4. See if you can work it out for yourself using the method outlined above.

**Quadratic functions can be written in the form:**

**y = ax2 + bx + c**

where a, b and c are constants and 'a' doesn't equal zero.

Quadratic graphs are always **parabolas **('U' shapes).

As they are curved, you need to plot a few more points than you do with the linear graphs. The U-shape could be anywhere on the graph depending on the values of a, b and c, so you may need to try a few values of x out until you have an idea of where it is.

Below we've done a table for the function **y = x2 - 4x - 5** taking values of x from -8 to 8:

x |
-8 | -6 | -4 | -2 | 0 | 2 | 4 | 6 | 8 |

y |
91 | 55 | 27 | 7 | -5 | -9 | -5 | 7 | 27 |

The really important bits of a quadratic are:

**Where it turns (the bottom of the 'U')**

**Where it crosses the x-axis (if it does!)**

So we just need to make sure that these are on our graph. Looking at our co-ordinates it appears that this happens somewhere between x = -2 and x = 6 so there's no need to go down as far as x = -8.

**All we need to do now is draw the axes, plot the points and draw a smooth U-shape through them:**

Now, if you've read the section on solving quadratics (see **Equations and Inequalities**) then you'll know that as well as factorising and using the quadratic formula, another way to solve them is by using the graph.

And the nice thing is it's really simple.

**The solutions of a quadratic are where the graph crosses the x-axis!**

This gives you the two x-values you need.

Now you can also see why some quadratics have 2 solutions (U-shape crosses the x-axis and comes back up), some have 1 solution (U-shape just touches the x-axis at a single point) and some have no solutions (U-shape is above the x-axis and doesn't cross it).

**You need to be able to:**

**1.** Plot and draw these.

**2.** Recognise the shapes.

**3.** Read the solutions from the graph (cubics only).

**Cubics** can be written in the form:

**y = ax ^{3} + bx^{2} + cx + d**

**Reciprocals** are where the x is on the bottom of a fraction.

The simplest cubic is y = x^{3}

**You can see below what it looks like:**

**The simplest reciprocal is:**

and it looks like this. It never touches the axes but gets closer and closer.The axes are known as **asymptotes.**

To plot them, do the same thing as for cubics (remembering the different shapes of cubics and reciprocals).

**Table - Axes - Plot - Draw - Label**

Try and think of an easy way to remember this!

The **solutions **of a cubic are where it crosses the x-axis and it can have up to 3 like the one shown below that has the solutions x = -4, x = 2and x = 5.

As all simultaneous equations you come across at GCSE are **linear** (can both be written in the form y = mx + c) their graphs will be straight lines.

The solution (x-value and y-value) is where the straight lines intersect (cross each other).

So instead of solving them using Algebra you can read their solution straight from the graph!

**So, for the simultaneous equations:** x + y = 10

x + y = 10 |

y - 2x = 1 |

the solutions can be read from the graph below as x = 3, y = 7

In an **inequality,** if you imagine that the inequality sign was an '=' sign then you've got a **linear equation.** We told you earlier that if you draw the graphs of a linear equation you get a straight line.

The only difference now is that the graph of the inequality includes the whole area (or region) on **one ** side of the straight line which you can shade in with your pencil.

**To find out which side of the straight line you need:**

**1.** Choose any point on the graph but not on the line as a **test point** (the origin is usually a good one unless it is on the line).

**2.** Put the x and y values of the point into the inequality. If the inequality works (is true - for example 2 > 7 would not be true) then that's the side of the line you want. If not then you want the other side.

Here's an example using the inequality

**2x + y > 7.**

First, make a table of values to draw the line

**2x + y = 7.**

* Remember:* you only need 4 or 5 values of x.

**When** x = -4, y = 15

**When** x = -2, y = 11

**When** x = 0, y = 7

**When** x = 2, y = 3

**When** x = 4, y = -1

X |
-4 | -2 | 0 | 2 | 4 |

Y |
15 | 11 | 7 | 3 | -1 |

Now plot the points and draw the line - **make it dotted as the inequality is 'greater than' and not 'greater than or equal to'.**

Choose a test point. We are going to choose (0, 0).

Put the x = 0 and y = 0 values into the inequality 2x + y > 7 to give 0 > 7 which is clearly not true so we want the other side of the line (see below):