FOUNDATIONS 07
Coordinates and Graphs
Coordinates turn numbers into positions, while graphs turn mathematical relationships into shapes that we can interpret visually.
The main idea
An equation tells us a relationship symbolically. A graph shows the same relationship geometrically.
\[\boxed{\text{equation}\longrightarrow\text{coordinate pairs}\longrightarrow\text{graph}\longrightarrow\text{behaviour}}\]From a graph we can see direction, steepness, intersections, maxima, minima and changing slope. These ideas lead directly into calculus.
1. The coordinate plane
The horizontal axis is the \(x\)-axis and the vertical axis is the \(y\)-axis. They meet at the origin \((0,0)\).
A point \((x,y)\) tells us first how far to move horizontally and then how far vertically.
2. Ordered pairs and quadrants
The order matters: \((2,5)\) and \((5,2)\) are different points.
The axes divide the plane into four quadrants. Signs of \(x\) and \(y\) tell us which quadrant contains a point.
3. From an equation to a graph
For \(y=2x+1\), each value of \(x\) determines one value of \(y\).
\[x=-1,0,1,2\quad\Rightarrow\quad y=-1,1,3,5.\]Plotting all such coordinate pairs produces a straight line.
4. Intercepts
The \(y\)-intercept occurs where \(x=0\). The \(x\)-intercept occurs where \(y=0\).
For \(y=2x+1\), the intercepts are \((0,1)\) and \((-\tfrac12,0)\).
\(y=0\Rightarrow2x+1=0\Rightarrow x=-\frac12\)
5. Gradient measures steepness and direction
For a straight line, gradient is the change in vertical position divided by the change in horizontal position:
\[m=\frac{\Delta y}{\Delta x}.\]A positive gradient rises from left to right; a negative gradient falls.
6. Curves have changing gradient
The graph \(y=x^2\) is not a straight line. Its steepness changes continuously.
On the left it falls, at the bottom it is momentarily horizontal, and on the right it rises. This leads to the central calculus question: what is the gradient at one exact point on a curve?
7. Intersections solve equations visually
When two graphs intersect, both equations have the same \(x\) and \(y\) values there.
Thus intersections provide a geometric way to solve simultaneous equations.
8. Graphs describe real change
If the horizontal axis represents time, the vertical axis can represent temperature, population, distance, infection prevalence or another changing quantity.
The graph then becomes a visual history of the system.
Coordinates locate points, equations generate sets of points, and graphs reveal the geometry of mathematical relationships. Gradient describes direction and steepness, intercepts show where graphs meet the axes, and intersections show where different relationships have the same values.
Why this matters for calculus
Calculus turns graphical ideas into precise mathematics. The derivative measures the instantaneous gradient of a curve; stationary points describe local maxima and minima; and the integral measures accumulated change and signed area. Learning to read a graph is therefore an essential preparation for calculus.