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Algebra 1298Updated Oct 3, 20264 pages

Understanding Math Functions

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Math formulas are the essential building blocks you need to solve problems across different areas of mathematics. This guide provides key formulas for geometry, algebra, and more that will help you solve problems quickly and accurately.
Math Function Notes – page 1

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Essential Math Formulas

Ever wonder how to find the distance between two points or figure out a parabola's vertex? These formulas are your toolkit for solving coordinate geometry and algebra problems.

The distance formula helps you find how far apart two points are: √(x2−x1)2+(y2−y1)2(x₂-x₁)²+(y₂-y₁)². When you need the point exactly between two locations, use the midpoint formula: (x1+x2)/2,(y1+y2)/2(x₁+x₂)/2, (y₁+y₂)/2.

For lines, the slope formula (rise/run) is y2−y1y₂-y₁/x2−x1x₂-x₁. You can write equations using point-slope form y−y1=m(x−x1)y-y₁=m(x-x₁) or the simpler slope-intercept form y=mx+by=mx+b. When dealing with parabolas, the quadratic formula −b±√(b2−4ac)-b±√(b²-4ac)/2a solves for the x-intercepts of y=ax²+bx+c.

Pro Tip: To find where a parabola reaches its highest or lowest point (the vertex), use x=-b/2a, then plug that x-value into the original equation to find y.

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Area and Perimeter Formulas

Need to calculate how much fencing for a yard or paint for a wall? These formulas will help you find the space inside (area) and distance around (perimeter) different shapes.

For squares and rectangles, area is straightforward: square area is s² (side squared), while rectangle area is l×w (length times width). Triangle area is ½(b×h) – half the base times the height. For circles, area equals πr² (pi times radius squared).

When calculating perimeters, square perimeter equals 4s (four sides), rectangle perimeter is 2l+2w (twice length plus twice width), and triangle perimeter is the sum of all three sides. The distance around a circle, called circumference, equals 2πr.

Remember: For irregular shapes like trapezoids area=h(b1+b2)/2area = h(b₁+b₂)/2, break them down into simpler shapes you already know how to calculate!

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Volume and Surface Area

When you jump into 3D shapes, you'll need formulas to find how much space is inside (volume) and how much material covers the outside (surface area).

A sphere's volume is 4/34/3πr³, while a cone's volume is 1/31/3πr²h. For containers like cylinders, volume equals πr²h, and a rectangular prism's volume is simply l×w×h (length × width × height).

Surface area calculations help you determine how much wrapping paper you'd need for a box or paint for a tank. A sphere's surface area is 4πr², a cylinder's surface area is 2πrh+2πr² (curved side plus two circular ends), and a rectangular prism's surface area is 2(wl+hl+hw).

Visualization Tip: Think of volume as how many cubes fit inside a shape, while surface area is like unfolding the shape and measuring its "skin."

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Logarithmic and Exponential Formulas

Logarithms and exponentials might seem tricky at first, but they're super useful for problems involving growth, decay, and solving certain equations.

The log product rule (logₐ(P·Q) = logₐP + logₐQ) lets you split a logarithm of multiplied numbers. When dividing inside a logarithm, use the log quotient rule: logₐ(P/Q) = logₐP - logₐQ. For exponents inside logarithms, the log power rule states logₐPqP^q = q·logₐP.

With exponentials, remember that multiplying the same base with different exponents means you add the exponents: a^m·a^n = a^m+nm+n. When dividing exponentials with the same base, subtract the exponents: a^m/a^n = a^m−nm-n.

Real-World Connection: These formulas are crucial for understanding compound interest, population growth, and radioactive decay. For example, the amount in a bank account with compound interest can be calculated using P1+r/n1+r/n^(nt).

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