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Formulas You’ll Need for the Accuplacer Next Generation Advanced Algebra and Functions Test

The following chart will give you the basic formulas you’ll need to ace the Advanced Algebra and Functions section of the Accuplacer Next Generation Test. You probably won’t be able to use this chart during this test, but it is an awesome tool for you to have an overview of the wide range of topics covered in this section. These are the very basic equations, so feel free to also check out our more complete study guides, and solve our sample problems to hone your skills at Union Test Prep!

Linear Equations

Formula Symbols Comments
\(A\cdot x + B\cdot y = C\) \(A, B, C = \text{any real number}\)
\(y= \text{dependent variable}\)
\(x = \text{independent variable}\)
Standard Form
\(y=m \cdot x + b\) \(y = \text{dependent variable}\)
\(m= \text{slope}\)
\(x = \text{independent variable}\)
\(b = y \text{-intercept}\)
Slope-Intercept Form. Try to convert linear equations to this format.
\(m = \dfrac{y_2 - y_1}{x_2 - x_1}\) \(m = \text{slope}\)
\(y_n = \text{dependent variable (point n)}\)
\(x_n = \text{independent variable (point n)}\)
This is a rearranged version of the point-slope form.
\(y-y_1 = m(x-x_1)\) \(y= \text{dependent variable}\)
\(x = \text{independent variable}\)
\(y_1 = y \text{ value of a point on the line}\)
\(x_1 = x \text{ value of a point on the line}\)
\(m = \text{slope}\)
Point-Slope form
\(x+a = b \Rightarrow x = b-a\)
\(x-a = b \Rightarrow x = b+a\)
\(x \cdot a = b \Rightarrow x = b \div a\)
\(x \div a = b \Rightarrow x = b \cdot a\)
\(x^a = b \Rightarrow x = \sqrt[a]{b}\)
\(\sqrt[a]{x} = b \Rightarrow x = b^a\)
\(a^x = b \Rightarrow x = \dfrac{\log b}{\log a}\)
\(a, b = \text{constants}\)
\(x = \text{variable}\)

Quadratic Equations

Formula Symbols Comments
\(x= \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) \(a,b = \text{constants}\)
\(c = \text{constant (y-intercept)}\)
\(x = \text{variable}\)
Quadratic Formula for equation in the form \(ax^2+bx+c=0\)
\((a \pm b)^2 = a^2 \pm 2ab + b^2\) \(a,b = \text{constants or variables}\) Square of a sum or difference
\(a^2-b^2 = (a+b)\cdot (a-b)\) \(a,b = \text{constants or variables}\) Difference of squares

Cubic Equations

Formula Symbols Comments
\(a^3-b^3 = (a-b) \cdot (a^2+ab + b^2)\) \(a,b = \text{constants or variables}\) Difference of cubes
\(a^3+b^3 = (a+b) \cdot (a^2-ab + b^2)\) \(a,b = \text{constants or variables}\) Sum of cubes

Geometry Equations

Formula Symbols Comments
\(A = s^2\) \(A = \text{area of a square}\)
\(s = \text{side length}\)
\(A = l \cdot w\) \(A = \text{area of a rectangle}\)
\(l = \text{length}\)
\(w = \text{width}\)
\(A = \dfrac{1}{2} b \cdot h\) \(A = \text{area of a triangle}\)
\(b= \text{base}\)
\(h = \text{height}\)
\(A = \pi \cdot r^2\) \(A = \text{area of a circle}\)
\(r = \text{radius}\)
\(A = h \cdot \dfrac{b_1+b_2}{2}\) \(A = \text{area of a trapezoid}\)
\(b_n = \text{base }n\)
\(h = \text{height}\)
\(C= 2 \pi r = \pi d\) \(C = \text{perimeter of a circle}\)
\(r = \text{radius}\)
\(d = \text{diameter}\)
\(V = s^3\) \(V = \text{volume of a cube}\)
\(s = \text{side length}\)
\(V = l \cdot w \cdot h\) \(V = \text{volume of a rectangular prism}\)
\(l = \text{length}\)
\(w = \text{width}\)
\(h = \text{height}\)
\(V = \dfrac{4}{3} \pi r^3\) \(V = \text{volume of a sphere}\)
\(r = \text{radius}\)
\(V = \pi r^2 h\) \(V = \text{volume of a cylinder}\)
\(r = \text{radius of base}\)
\(h = \text{height}\)
\(V = \dfrac{1}{3} \pi r^2 h\) \(V = \text{volume of a cone}\)
\(r = \text{radius}\)
\(h = \text{height}\)
\(V = \dfrac{1}{3} l \cdot w \cdot h\) \(V = \text{volume of a pyramid}\)
\(l = \text{length}\)
\(w = \text{width}\)
\(h = \text{height}\)
\(d= \sqrt{\mathstrut (y_2 - y_1)^2 + (x_2-x_1)^2}\) \(d = \text{distance between two points}\)
\(y_n = y \text{ value at point n}\)
\(x_n = x \text{ value at point n}\)
\(a^2 + b^ 2 = c^ 2\) \(a,b = \text{legs of a right triangle}\)
\(c = \text{hypotenuse of a right triangle}\)
Pythagorean theorem
\((x-h)^2 + (y-k)^2 = r^2\) \((h,k) = \text{center of a circle}\)
\(r = \text{radius}\)
Standard form of a circle
\(x^2 + y^2 + Ax + By + C = 0\) \(x, y = \text{variables}\)
\(A,B,C = \text{constants}\)
General form of a circle

Trigonometry Equations

Formula Symbols Comments
\(\sin^2 \theta + \cos^2 \theta = 1\)   Pythagorean Identity
\(\sin 2\theta = 2 \sin \theta \cdot \cos \theta\)
\(\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta -1\)
\(\tan 2\theta = \frac{2 \tan \theta}{1-\tan^2 \theta}\)
  Double Angle formulas

Formulas with Diagrams


\[\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\csc \theta} \\ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sec \theta} \\ \tan \theta = \frac{\sin \theta}{\cos \theta} =\frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\cot \theta}\]


\[\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} \\ a^2 = b^2 + c^2 -2bc \cos A \\ b^2 = a^2 + c^2 -2ac \cos B \\ c^2 = a^2 + b^2 -2ab \cos C\]

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