what has algebra got to do with geometry?
We have learned that Algebra is an area of mathematics that uses variables in the form of letters and symbols, in place of quantities or numbers in formulae and equations. We also know that Geometry is an area in mathematics that studies points, lines, multi-dimensional objects and shapes, surfaces and solids. But are they so very different from each other??
Coordinate Geometry:
One way that algebra and geometry can be related is through the use of equations in graphs. We can plot a set of points (x, y) according to an equation (for example, the line graph on the left!) to form a graph. That's one way that algebra is related to geometry. A set of points can satisfy any equation which can produce any type of graph, not just straight lines.
Geometric concepts:
Let's think about all the two-dimensional shapes we know - to find the area of any of these shapes we use their specific area formula. We have to substitute the corresponding dimensions into the area formula and solve for the area. Sometimes we also use the formulae to find missing dimensions by rearranging the formula. We need to use a lot of our algebraic skills in this area!
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the pythagorean theorem
transformations and functions:
In geometry, a transformation is a way to change the position of a figure or shape. Below are the four main types of transformations:
So how is transformation, a geometric concept, related to functions, an algebraic concept? Well, first of all, what is a function? A function is like a machine, it performs a function (some know it as rules) on an input and produces a corresponding output. In algebra, many equations are actually functions because when we substitute one value into an equation, we get another value. As we have seen in Coordinate Geometry at the top, we can graph these equations or functions. After we graph the equations, we can move the graph up or down or side to side by adding a constant (value) either to x or y in the function. See below for some illustrations of how functions can "transform":
In our next unit, we will look at functions in more detail!