Transforming Functions

Questions involving reflecting or shifting graphs stump a great many students.

But this needn’t be! 

Four simple rules govern all transformation questions encountered on the SAT/ACT. Master these laws, and all such questions suddenly become easy ones.

Listed below is what you need to know.

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Horizontal Reflection (across the y-axis)

Replace x with -x. 

For example: If f(x) = x^2– x+1, the horizontal reflection is f(x) = (-x)^2–(-x)+1 = x^2+x+1.

Vertical Reflection (across the x-axis)

Replace y with -y.
For example: If g(x) = 3x–2 i.e. y = 3x–2, the vertical reflection is (-y) = 3x–2 and y = -3x+2. Therefore, g(x) = -3x+2.

Horizontal Shift, h units

Replace x with x–h.
For example: If f(x) = x^2–x is shifted 4 units left, h = -4, h–k = h–(-4) = h+4, and the shifted function is f(x) = (x+4)^2–(x+4) = x^2+8x+16–x–4 = x^2+7x+12.

Vertical Shift, k units

Replace y with y-k (or simply add k to the function).
For example: If y = |6x–1| is shifted 3 units up, k = 3, y–k = y–3, and the shifted function is y–3 = |6x–1|. Therefore, y |6x–1|+3.

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For practice, search Google for worksheets covering any or all topics listed above, pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Geometry You Need to Know

For more than 2500 years, since the time of Euclid, geometry has occupied a central place in the study of mathematics, and these problems form an important subset of questions encountered on the SAT/ACT. 

Luckily, the particular facts and concepts you need to know are few and easy to review. 

A comprehensive list of these elements follows. Make sure you’ve mastered each one.

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Perimeter (with radius r and diameter d)
Polygons: Add all sides.
Circumference of a circle = 2πr or πd

Area formulas (with base b and height h)

Triangle = bh/2.

Parallelogram: bh (includes rectangles and squares)

Trapezoid = h(b1+b2)/2.

Circle = πr^2.

Volume formulas

Rectangular prisms = Bh, where B = rectangular base area and h = height of the object (includes boxes, including cubes), 

Right cylinders = Bh, where B = circular base area and h = height of the object.

Right cones = (1/3)Bh, where B = circular base area and h = height of the object.

Famous figures
See study sheet here.

Triangle inequality theorem
The length of any side in a triangle must be between the sum and difference of the other two sides.

For example: The lengths 8, 10, and 2 could not form a triangle since 2 is not between 2 and 18.

Pythagorean Theorem: a^2+b^2 = c^2 (where, for any right triangle, a and b are legs and c is the hypotenuse

Apply the Pythagorean Theorem to find missing sides in a right triangle.

Use key right triangle “triples” (3x : 4x : 5x, 5x : 12x : 13x) to find missing sides in a right triangle.

Use ratios of sides in special right triangles (30-60-90 = x : x√3 : 2x, 45-45-90 = x : x : x√2) to find missing sides in a right triangle.

Parallel lines are cut by a transversal
Know how to use "big angles" and "small angles" formed to find measures of unknown angles in figures.

Regular hexagon
A regular hexagon can be divided into six equilateral triangles by drawing segments between opposite vertices. Each equilateral triangle can then split into two 30-60-90 triangles, from which various lengths can be inferred.

Questions involving circles and radii
Circle problems can often be solved by drawing radii to indicated points on the circle and noting that all radii have the same length.

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For practice, search Google for worksheets covering any or all of the above, pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.