Academic Decathlon Math Learner

Shortcut 1: Using the Calculator's Equation Solver

Whenever you need to find the solution(s) for an equation quickly, your calculator's equation solver will come in handy. Here is how to use the Ti-84 Plus CE's built in equation solver:

• Press the MATH key, and then press up the up arrow.
• The cursor should be hovering over "Numeric Solver". Press enter.
• You will now see two boxes. One will say E1, and one will say E2. Enter the left hand side on E1 and the right hand side on E2.
• Press the GRAPH key twice to see the solution next to where it says "X=". If there is no solution, the calculator will tell you.

Now, if you need to check a specific solution, simply change the guess value(the number next to "X=") to the number you want to check and press the graph key again. If the same number comes up next to X=, then the solution is valid.

Shortcut 2: Evaluate a Function Using Your Calculator

When you need to find the value of a function at a specific point, you can do the following:

• Press the Y=. You will see a bunch of lines saying Y1=, Y2=, etc.
• Enter your function in one of the Y= lines, using the variable X from the button that says X,T,θ,n.
• If you want to find the value of the function at, say x = 2, you simply type in Y1(2), assuming Y1 is the line where you entered your function.
• To do this, press the ALPHA and then TRACE. You will get the option to choose Y1, Y2, etc.
• Click on the corresponding number, and next to that, type in parentheses around the value you want to check.
• Press enter, and the calculator will give you the value of the function at that point.

Important Concepts

Some concepts that are nice to know:

• Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a, where x is the solution to the quadratic equation ax² + bx + c = 0
• Discriminant: b² - 4ac, which determines the nature of the roots of a quadratic equation.
• If b^2 - 4ac < 0, then the quadratic has no real solutions. If b^2 - 4ac = 0, then the quadratic has exactly one real solution. If b^2 - 4ac > 0, then the quadratic has two distinct real solutions.

Important Concepts

Some concepts that are nice to know:

• Adding/Subtracting polynomials involve combining like terms.
• Rational Root Theorem: If a polynomial has rational roots, they can be expressed as p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
• Factor Theorem: A polynomial f(x) has a factor (x - c) if and only if f(c) = 0.
• Remainder Theorem: When a polynomial f(x) is divided by (x - c), the remainder is f(c).
• Function Transformations:
• Vertical shifts: f(x) + k shifts the graph vertically by k units.
• Horizontal shifts: f(x - h) shifts the graph horizontally by h units.
• Vertical stretches/compressions: a * f(x) stretches the graph vertically by a factor of |a|.
• Reflections: -f(x) reflects the graph across the x-axis and f(-x) reflects it across the y-axis.
• Horizontal stretches/compressions: f(bx) compresses the graph horizontally by a factor of |b|.
• Vertical stretches/compressions: f(x) compresses the graph vertically by a factor of |a|.
• Inverses: f(x) and f^(-1)(x) are inverse functions if and only if f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.
• To quickly find inverses, substitute a point (a, b) into f(x) and check that substituting b into another function yields a.
• Domain and Range of Radical, Rational, Logarithmic, and Exponential Functions:
• Radical: The domain is all x such that the radicand is non-negative. The range is all y such that y is non-negative.
• Rational: The domain excludes values that make the denominator zero. The range can be determined by analyzing the behavior of the function.
• Logarithmic: The domain is all x such that the expression inside the log is positive. The range is all real numbers.
• Exponential: The domain is all real numbers. The range is all y > vertical shift.
• Note that the domain and range can be affected by transformations applied to the parent functions.

Important Concepts

Some concepts that are nice to know:

• Pythagorean Theorem: a² + b² = c², where c is the hypotenuse of a right triangle.
• Distance Formula: d = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are two points in the plane.
• Midpoint Formula: M = ((x1 + x2)/2, (y1 + y2)/2), which gives the midpoint between two points.
• Perpendicular Bisector: The perpendicular bisector of a segment is the line that is perpendicular to the segment and passes through its midpoint.
• Equation of a Circle: The equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².
• The diagonals of a parallelogram bisect each other.
• The three perpendicular bisectors are concurrent(intersect at one point)
• The three medians are concurrent at a point where it divides each median in a 2:1 ratio.
• The concurrency point of the medians is the average of the coordinates of the vertices.

Important Concepts

Some concepts that are nice to know:

• Sine, Cosine, and Tangent Ratios: In a right triangle, sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent.
• Law of Sines: a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the lengths of the sides opposite angles A, B, and C respectively.
• Law of Cosines: c² = a² + b² - 2ab * cos(C), which relates the lengths of the sides of a triangle to the cosine of one of its angles.
• Graphing Trigonometric Functions:
• Amplitude: The coefficient in front of the sine or cosine function, which affects the height of the waves.
• Period: 2π divided by the coefficient of x inside the function.
• Trigonometric Identities: Some common identities include:
• sin²(θ) + cos²(θ) = 1
• tan(θ) = sin(θ)/cos(θ)
• Double Angle Identities:
• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) - sin²(θ)
• tan(2θ) = 2tan(θ)/(1 - tan²(θ))
• Half Angle Identities:
• sin(θ/2) = ±√((1 - cos(θ))/2)
• cos(θ/2) = ±√((1 + cos(θ))/2)
• tan(θ/2) = ±sin(θ)/(1 + cos(θ))
• cot(θ/2) = ±(1 + cos(θ))/sin(θ)
• Sum and Difference Identities:
• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
• cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
• cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
• tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B))
• tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B))
• Sum-to-Product Identities:
• sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2)
• sin(A) - sin(B) = 2cos((A + B)/2)sin((A - B)/2)
• cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2)
• cos(A) - cos(B) = -2sin((A + B)/2)sin((A - B)/2)
• tan(A) ± tan(B) = sin(A ± B)/(cos(A)cos(B))
• cot(A) ± cot(B) = sin(B ± A)/(sin(A)sin(B))
• Product-to-Sum Identities:
• sin(A)sin(B) = 1/2[cos(A - B) - cos(A + B)]
• cos(A)cos(B) = 1/2[cos(A - B) + cos(A + B)]
• sin(A)cos(B) = 1/2[sin(A + B) + sin(A - B)]
• Inverse Trigonometric Functions:
• sin⁻¹(x) or arcsin(x) is the angle whose sine is x.
• cos⁻¹(x) or arccos(x) is the angle whose cosine is x.
• tan⁻¹(x) or arctan(x) is the angle whose tangent is x.
• Domain and Range of Trigonometric and Inverse Functions:
• Domain of sin(x) and cos(x): all real numbers
• Range of sin(x): [-1, 1]
• Range of cos(x): [-1, 1]
• Domain of tan(x): all real numbers except (π/2 + nπ), where n is an integer
• Range of tan(x): all real numbers
• Domain of sin⁻¹(x): [-1, 1]
• Range of sin⁻¹(x): [-π/2, π/2]
• Domain of cos⁻¹(x): [-1, 1]
• Range of cos⁻¹(x): [0, π]
• Domain of tan⁻¹(x): all real numbers
• Range of tan⁻¹(x): (-π/2, π/2)