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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. - 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 s...

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. - 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 s...

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. - 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 s...

For well over 2000 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. - 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 inequ...

For well over 2000 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. - 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 inequ...