Graphing and Geometry for the DAT
/Learn key DAT concepts about graphing and geometry, plus practice questions and answers
Table of Contents
Part 1: Introduction to graphing and geometry
Part 2: Functions
Part 3: Graphing
a) Linear functions
b) Parabolic functions
c) Inequalities
Part 4: Geometry
a) Shapes
b) Trigonometry
c) Special right triangles
Part 5: Helpful Terms
Part 6 Questions and answers
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Part 1: Introduction to graphing and geometry
Graphing functions is an integral part of mathematics, and is likely a topic you learned as early as middle school. Geometry is another subject that you’ve likely already been exposed to. Regardless of the last time you took a math class, you can do well on the QR section. This guide will help you remember or re-learn everything you need to know about functions, graphing, geometry, and trigonometry for the DAT. Pay attention to any bolded terms, and use the practice questions at the end to test your knowledge.
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Part 2: Functions
Functions are an important concept to understand for the DAT and will set the table for our discussion on graphing. A function is a mathematical equation that produces one output from a given input. For the DAT, functions take the form of \( f(x)=x \)
The left side of the function, \( f(x) \), represents the input value. For example, \( f(2) \) would mean 2 is being input into the function. The right side of the function is the output. If given the function \( f(x)=x^2 \) and the input is 3, then the output is \( 3^2=9 \).
A common type of question that shows up on the QR section is solving composite functions. A composite function is a function whose value is determined from two or more other functions. Composite functions usually follow the form of \( f∘g \), where f represents the first function, g represents the second function, and the open circle is a composition operator. This statement can read as f of g of x, and it can also be written as \( f(g(x)) \). Let’s look at a few example problems.
Find \( f∘g \)
$$f(x)=4x+3$$
$$g(x)=x^3-12$$
To solve this problem, start by rewriting f of x. It can be rewritten as follows:
$$f(g(x))$$
Now, replace g of x with its value.
$$f(x^3-12)$$
Let’s look at the original function \( f \). We are given \( f(x)=4x+3 \), and now we have \( f[x^3-12] \). To solve this problem, simply plug \( x^3-12 \) into the function \( f(x) \).
$$f(x^3-12)=4(x^3-12)+3$$
Distribute the 4, combine like terms, and you’re done.
$$f∘g=4x^3-45$$
Composite functions can also take the form of \( g∘f \). These problems are solved the same way the previous problem was, but the functions are swapped. Here is an example.
Find \( g∘f\)
$$f(x)=x+3$$
$$g(x)=x^2-2$$
Similar to the previous problem, we can rewrite \( g∘f \), this time as \( g(f(x)) \). Plug in \( f(x) \)and solve as follows:
$$g(x+3)=(x+3)2-2$$
$$=x^2+6x+9-2$$
$$g∘f=x^2+6x+7$$
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