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Various mathematical concepts from term 1, including calculating slope, distance formula, solving for the slope of perpendicular lines, point slope form, slope intercept, equation of a circle, completing the square, parabolas, ellipses, hyperbolas, radians to degrees and vice versa, trigonometric equations with circles, trigonometric functions, trigonometric identities, even and odd functions, transformations, linear functions, polynomial functions, power functions, and root functions. It also includes definitions and instructions for solving problems related to these concepts.
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The slope of a nonvertical line is the change in y over the change in x The slope of a vertical line is not defined. TERM 2
DEFINITION 2 can be replaced by drawing graph and using the pythagorean theorem TERM 3
DEFINITION 3 take the negative reciprocal (flip the fraction) TERM 4
DEFINITION 4 yy1= m(xx1) TERM 5
DEFINITION 5 y = mx + b
(x - h)2 + (y-k)2 = r2f the equation of a circle is in the standard form, we can easily identify the center of the circle, (h, k), and the radius, r. Note: The radius, r, is always positive.In an equation, if the sign preceding h and k , ( h, k) are negative, then h and k are positive. TERM 7
DEFINITION 7 Move the loose number over to the other side.Take half of the x- term (that is, divide it by two) (and don't forget the sign!), and square it. Add this square to both sides of the equation. completing-the-square animationConvert the left-hand side to squared form. Simplify the right-hand side. Square-root both sides. Remember to do "" on the right-hand side. Solve for "x =". Remember that the "" gives you two solutions. Simplify as necessary. TERM 8
DEFINITION 8 Here we regard a parabola as a graph of an equation of the form y=ax^2+bx+cThe graph of an equation is symmetric with respect to the y-axis if the equation is unchanged when x is replaced by -x. It flips over the x axis if negative and open upside downThe graph of an equation is symmetric with respect to the x-axis if the equation is unchanged when y is replaced by -y. TERM 9
DEFINITION 9 The x-intercepts of a graph are the x-coordinates of the points where the graph intersects the x-axis. They are found by setting y = 0 in the equation of the graph.The y-intercepts are the y-coordinates of the points where the graph intersects the y-axis. They are found by setting x = 0 in its equation. TERM 10
DEFINITION 10 x^2/a^2 y^2/b^2 = 1
If f satisfies f (-x)= -f(x) for every number x in its domain, then f is called an odd function. if it is inconsistent then it is neither even nor odd. TERM 17
DEFINITION 17 -f (x) reflects f (x) over the x-axis.f (-x) reflects f (x) over the y- axisf (x + a) translates f (x) horizontally (a neg moves right and pos left)f (x)+ a translates f (x) verticallyf (ax) stretches/compresses f (x) horizontally (if whole number the graph gets taller and skinnier, if a fraction it gets wide and short)af(x) stretches/compresses f (x) vertically (if whole number the graph gets taller and skinnier, if a fraction it gets wide and short) TERM 18
DEFINITION 18 linear function of x, we mean that the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula for y =f(x)=mx +b where m is the slope of the line and b is the y-intercept. A characteristic feature of linear functions is that they grow at a constant rate. TERM 19
DEFINITION 19 A polynomial is an expression of one variable in the form a n x n + a n-1 x n-1 + ... + a 2 x 2 + a 1 x + a 0 , where a 0, a 1,, a n are real numbers with a n 0 and n is a positive integer. Many real-life situations are easily modeled by polynomial functions. Some of the most familiar are quadratic functions, which are functions of the form f (x) = ax 2 + bx + c TERM 20
DEFINITION 20 A function of the form f (x)= x^a, where a is a constant, is called a power function.
f (x)= x^aa = 1/n , where n is a positive integerThe function f(x) = x^1/n = the nth root of x is a root function. TERM 22
DEFINITION 22 The exponential functions are the functions of the form f(x) =a^x, where the base a is a positive constant.
