Graphs are just "pictures" of equations
Graphs are a way of visualising what an equation tells us about the relationship between x and y
You have already seen how to turn an equation into a graph:
Substitute the x values into the equation to find the corresponding y values
-
Plot these points on the graph
-
Join up points
-
Example:
Different types of equations make different types of graphs
We can put equations into groups based on their characteristics.
There are lots of different types of equations, but so far we have focussed on two.
Linear equations
1)
We can recognise a linear equation because it doesn't involve any powers of x (no x2, x3etc.)
y=3x+9 y =
-
8x+7 y=
-
12x
These are all examples of linear equations - we can see this easily because there are no powers of x involved.
When we turn a linear equation into a graph using the table method, it will always make a linear graph.
A linear graph is always a straight
line
-
that's where the name
line
ar comes from.
Quadratic equations2)
We can recognise a quadratic equation because it always includes x
2
y=x
2
+4x+9 y = x
2
-
27 y= 12x
2
These are all examples of quadratic equations.
Note: quadratic equations always include x
2
. And in a quadratic equation, x
2
is always the highest power (which means that a
quadratic equation will never have x
3
, or x
4
etc. Equations which do involve those powers are not quadratic
-
they have other
names).
When we turn a quadratic equation into a graph using the table method, it will always make a quadratic graph.
A linear graph is always a curve shape
-
some people call it a
horseshoe shape
or a U shape (although it can also sometimes
be upside down
∩
)
IMPORTANT: When we plot a quadratic graph we don't join up our points using a ruler
-
we do it free hand and try to draw
a smooth curve.
Finding x and y Intercepts of Quadratic Graphs
Remember the x intercepts are the points where the graph crosses through the x axis.
The y intercept is the point where the graph crosses through the y axis.
If we can see the graph we can easily find the intercepts
-
we just read them off the graph.
But sometimes we are just given the equation of the graph
-
we can't see the graph itself.
To find the intercepts we
could
plot the graph. But this is quite a long process.
There is a quicker way to find the intercepts using
algebra
Example: Find the x and y intercepts of the graph with the equation y = x
2
+ 4x
-
12
Finding the y intercept:
This is pretty straightforward. We just look at our equation and find the "constant" (this is the official name for the number
by itself - the one that isn't attached to x, or x2).
In this equation, the constant is -12.
So the y
-
intercept is
-
12
Examples of Quadratic Graphs
Graphs and Equations
-
Key points from the lesson
01 November 2025 20:41
New Section 78 Page 1
