Senior Math Section 6-1 Notes Rectangular Coordinates and Lines Label the following 1. quadrant 1 2. quadrant 2 3. quadrant 3 4. quadrant 4 5. origin 6. x-axis 7. y-axis 8. Ordered Pair (x, y) at (2, 1) x is also called the abscissa y is also called the ordinate "If you can create a t-table, you can graph anything!" Domain and Range Relation: a set of ordered pairs Function: a relation in which each x value maps to exactly 1 y value. Domain: the x values of the relation (without repeat) Range: the y values of the relation (without repeat) Functions Vertical Line Test: if a vertical line can hit the graph of a relation more than once it fails the vertical line test and is not a function. Examples Graphically Function? State Domain and Range. Functions describe a relationship between two variables and gives this relationship a name example: f(x) = 2x x is the independent variable, y is the dependent variable To evaluate a function, replace the x with what is inside the parenthesis When graphing functions, treat f(x) as y.
Senior Math Section 6-2&3 Notes Intercepts: Where the graph hits the x-axis or y-axis to find an x-intercept, plug in 0 for y to find a y-intercept, plug in 0 for x example Slope Slope: the rate of change of a function examples: Types of slope positive negative zero undefined slope equation rise run y - y x - x
Linear function: Creates a straight line. The x has power 1. Equation of a line in slope-intercept form y = mx + b Function Notation: If you see f(x) = 2x + 1, you can rewrite it as y = 2x + 1 Graphing an equation in y=mx + b form 1. put a point at the y-intercept 2. use the slope (rise/run) to find more points Example #45: given the point (-3, 0) and a slope of -3, complete the following: 1. graph the line 2. state the equation 3. state the intercepts x: y: 4. is (3, -3) on the line? 5. point (b, -3) is on the line, find b 6. point (3, a) is on the line, find a 7. find x if (x, 2x) is on the line
Notes on regression and correlation Regression line/equation - also called the line of best fit - the line that best fits a series of data points - used to make predictions Example: construct a best-fit line, state the equation, and predict the y value when x is 20. Correlation: - describes the strength of a relationship on a scale from -1 to 1 - positive means that as x goes up, y goes up - negative means that as x goes up, y goes down value strength what it looks like -1 perfect negative 0 no relationship 1 perfect positive Add the number line below:
On a calculator: To show correlation, follow these commands 2 nd Catalog (on the zero) Scroll down to DiagnosticOn Press ENTER twice To calculate a regression equation 1. press STAT and select EDIT 2. clear any data in L1 and L2 and enter your data 3. press STAT, press the right arrow to select CALC, and choose press 4. LinReg(ax+b) 4. press ENTER to run the regression program Example: How far away from school do you live? How many minutes before 7:30 do you wake up? Higher Power Regression (for section 6-4) 1. Find the change between each y value (subtraction) and write them down. If the new numbers are the same, then the pattern is linear (slope). Run Linear Regression. 2. Find the change between the numbers you wrote down in the last step and write them down. If the new numbers are the same, then the pattern is quadratic. Run Quad Regression. 3. Repeat this process until the numbers you write down are the same. If it takes 3 steps, the highest power is 3, run trinomial regression.
Senior Math Section 6-4 Notes Example: identify a, b, and c for F(x) = x 2 3x + 1 Vertex The lowest (min) or highest (max) point on the parabola. Location: b b, f 2a 2a Vertex on a Calculator Type the equation in Y1 Graph and trace to the the left of the vertex 2 nd Calculate, select Minimum or Maximum Press enter to mark the left Scroll to the right of the vertex and press enter twice Quadratic Formula The quadratic formula Tells where the graph hits the x-axis Calculator Method Type the equation in Y1 Type 0 in Y2 Graph 2 nd Calculate, Intersection Scroll Close and press enter 3 times x = b ± b 2 4ac 2a
Derivative Supplemental Slope as Speed When a graph shows distance on the y-axis and time on the x-axis, the slope will show the speed. speed = distance / time Tangent Line A line that describes the slope at 1 point along a graph (see GSP demo) The slope of the tangent line tells the speed at that point (instantaneous velocity) Draw the following tangent lines and find the slope X = -2 X = 0 X = 1 What is a derivative? The slope of a tangent line. The tangent line is a picture of the slope. The purpose of knowing the slope of the tangent line is that it tells rate of change (speed!!). Notation o Original function f (x) y o First Derivative: f (x) y o Second Derivative: f (x) y The derivative of a number is 0 example: if f(x) = 2, then f (x) = 0 The derivative of a linear is the slope example: if f(x) = 5x + 2, then f (x) = 5
Power Rule (for polynomials) For each term, multiply the power by the coefficient in front and decrease the power by 1. Formal definition: if n is a rational number, then the function f(x) = xn is differentiable and is equal to: derivative of [ax n ] = anx n 1 Example: f(x) = 3x 3 4x 2 + x 6 f (x) = 3 3x 2 4 2x 1 + 1x 0 0 à f (x) = 9x 2 8x + 1 Examples: y = 2x 5 x 4 + x 1 y = To Find the Second Derivative Apply the power rule to the original to find the first derivative and then apply the power rule to the derivative to find the second derivative. The Physics of Derivatives The original function represents displacement over time (how far from the start) The first derivative represents velocity over time (how fast). Positive = moving forward. Negative = moving backwards. The second derivative represents acceleration over time (speeding/slowing). Positive = speeding up. Negative = slowing down. Example Find the displacement, velocity, and acceleration on the function below at time x = 3 seconds if f(x) = x^2 3
Section 6-5 and 6-6 Exponentials and Logarithms Exponentials The input is a power, the output/answer is a magnitude Basic Form of an exponential: f(x) = b x where b is the base and x is the power. You can find values by hand, from the graph, or from the table. e is a constant, like π, and is approximately equal to 2.7182 General Graph: Logarithms The input is a magnitude, the output/answer is a power. Basic Form of an exponential: f(x) = log b x where b is the base and x is the magnitude. You can find values by hand, from the graph, or from the table. The purpose of a Logarithm is to shrink huge numbers down into small numbers. General Graph: Exponentials and Logarithms are opposites How to change them from one to the other: y = loga x is the same as a y = x Loop the base!! Examples Rewrite using logs: 7 2 = 49 becomes 7 = log2 49
Graphing logs on the calculator: log B A = log A log B or log B A = ln A lnb graph f(x) = log3 (2x 1) y1 = log (2x- 1) / log 3