### SAT ACT math Rules – You must know before taking SAT/ACT Tests

SAT/ ACT Math Tutors in Dubai Abu Dhabi & Sharjah, We teach most important Rules along with Questions, sample of some common topics are shown below, also there are more specific topics by sub category.

- Word Problems – 60% of the ACT/SAT math is word problems. More, less, fewer, higher, lower, double, half, etc.
- Properties of Integers – 30% of Q’s used by the ACT/SAT are integers. Even, odd, positive, negative, consecutive, prime, factor, multiple, square, etc.
- Rate- Time : 1/a + 1/b = (a + b)/ab. 1/2 + 1/3 = (2+3)/(2*3)
- Sequences : Arithmetic = add/subtract the Common Difference. Geometric = multiply/divide the Common Ratio, ‘n’th Term of sequence and Sum of ‘ n’ terms
- Part+Part=Whole.
- Rates & Proportions.
- Increase, decrease, percent of a percent, Simple and Compound interest
- Algebraic Operations & solving equations. Combine like terms – watch out for negative signs. Easy – don’t miss these.
- Algebraic Inequalities – graph them quickly.
- Absolute Value.
- Exponents, roots, logarithms.
- Linear Functions. ax+by=c & y=mx+b
- Coordinate Geometry in the xy-plane.
- Systems of Equations
- Evaluate f(x).
- Quadratics, parabolas, factor and FOIL.
- Lines & Angles. l is parallel to m, l||m –> Same Slope
- Triangles, right, isosceles, 3-4-5 and 5-12-13.
- Quadrilaterals, squares, rectangles, trapezoids.
- Circles & sectors. They love the angle of a piece of a pie chart.
- Multiple Geometry Figures. Big Area – Small Area is usually the answer.
- Two SOHCAHTOA questions, usually straightforward.
- Average, Mean, Median, rarely mode.
- Probability
- Counting, Factorials, Permutations, & Combinations.

**The graph of the function f is the graph of the function g compressed horizontally by a factor of 2 and reflected over the y-axis. Which of the following correctly defines the function f?**

A) f(x) = g(-2x)

B) f(x) = -g(2x)

C) f(x) = 2 g(-x)

D) f(x) = -2 g(x)

we have to compress g horizontally by a factor of 2 and reflect it over the y-axis. Given what we’ve learned, this is pretty easy.

To compress the graph horizontally by a factor of 2, we replace x with 2x: —> g(2x)

To reflect the graph over the y-axis, we replace x with -x

$g(2(−x))=g(−2x)$

The answer is **A**.

To understand the above Example in better way, please find the 7- Steps Method of same topic:

**Step 1 ) f(x) = x^2 + 2x**

whose graph looks like

#### Step 2 ) Shifting up and down

To shift the graph up, add a constant at the end of the function. For example, f(x) + 2 = x^2 + 2x + 2 would shift the graph up 2 units.

To shift the graph down, subtract a constant at the end of the function. f(x) – 2 = x^2 + 2x – 2 would shift the graph down 2 units.

#### Step 3 ) Shifting left and right

To shift the graph to the left by 1 unit, replace x with x + 1:

f(x + 1) = (x + 1)^2 + 2(x + 1)

To shift the graph to the right by 1 unit, replace x with x – 1:

f(x – 1) = (x – 1)^2 + 2(x – 1)

The substitutions for left and right are counterintuitive to a lot of students because they’re the opposite of what you might expect.

#### Step 4 ) Reflecting across the x-axis

To reflect the graph across the x-axis (flip it upside down), multiply the function by -1:

-f(x) = -x^2 – 2x

#### Step 5 ) Reflecting across the $y$-axis

To reflect the graph across the y-axis, replace $x$ with $−x$:

f(-x) = (-x)^2 + 2(-x) = x^2 – 2x

#### Step 6) Stretching/Compressing vertically

To stretch the graph vertically by a factor of 2, multiply $f(x)$ by 2:

2f(x) = 2(x^2 + 2x) = 2x^2 + 4x

You can think of this as the graph getting “taller.”

To compress the graph vertically by a factor of 2, multiply f(x) by 1\2:

1\2 * f(x) = 1/2 * (x^2 + 2x) = 1/2 * (x^2) + x

Think of this as the graph getting “shorter.”

#### Step 7) Stretching/Compressing horizontally

To stretch the graph horizontally by a factor of 2, replace x with ( 1$x$:

f ((1/2)x ) = ((1/2 )x)^2 + 2 *(1/2 )x = 1/4 * (x ^2) + x

Notice that the right side doesn’t get stretched as much. That’s because the points on the right side of the graph have smaller $x$-values (absolute value-wise). Doubling a small number doesn’t do as much as doubling a large number.

To compress the graph horizontally by a factor of 2, replace $x$ with 2x:

f(2x) = (2x)^2 + 2(2x) = 4x^2 + 4x

By the way, these substitutions for horizontal stretch and compression are also counterintuitive to a lot of students. Be careful of horizontal transformations (shifting left and right, stretching/compressing). It’s often the opposite of what you might expect.

Notice that the right side doesn’t get compressed as much. That’s because the points on the right side of the graph have smaller $x$-values (absolute value-wise). Halving a small number doesn’t do as much as halving a large number.

#### Absolute Value

To reflect all the points on the graph with negative y-values across the x-axis, take the absolute value of the function:

$∣ $