Chapter 4 LINEAR FUNCTIONS

Section 4-1 LINEAR FUNCTION – A function whose graph forms a straight line. Linear functions can describe many real-world situations, such as distances traveled at a constant speed. LINEAR EQUATION – Any equation that can be written in standard form.

STANDARD FORM OF A LINEAR EQUATION Ax + By = C (where A, B, and C are real numbers and A and B are not both 0) X and y both have exponents of 1. X and y are not multiplied together. X and y do not appear in denominators, exponents, or radical signs.

Lesson 4-2 y-intercept: The y-coordinate of the point where the graph intersects the y-axis. The x-coordinate of this point is always 0. x-intercept: The x-coordinate of the point where the graph intersects the x-axis. The y-coordinate of this point is always 0.

Graphing Ax + By = C Using Intercepts Find the x-intercept by _______________ Find the y-intercept by________________ Graph the line by____________________ Find the x-intercept by letting y = 0 and solving for x. Find the y-intercept by letting x equal 0 and solving for y. Graph the line by plotting the points containing the intercepts and then connecting the points with a straight line.

Lesson 4-3 Rate of change – a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. change in dependent variable change in independent variable

Rise – the difference in the y-values of two points on a line. Run – the difference in the x-values of two points on a line. slope – The ratio of rise to run for any two points on the line. Slope = rise = change in y run change in x The slope of a horizontal line is 0. The slope of a vertical line is undefined.

Lesson 4-4 You can use the slope formula to find how quickly a quantity is changing. Slope = change in y change in x Slope Formula m = y2 – y1 x2 – x1

Lesson 4-5 Direct Variation – a special type of linear relationship that can be written in the form y = kx Constant of variation – “k” the ratio of y/x in an equation with direct variation

Lesson 4-6 Slope-intercept form y = mx + b, Where m is the slope, and b is the y intercept.

Lesson 4-7 Point-Slope Form . y – y1 = m(x – x1) This formula can be used when you are given a point (x1, y1) and the slope (m), or two points (x1, y1), (x2,y2) When simplified, point- slope form becomes slope intercept form and can be used to graph a line.

Forms of Linear Equations STANDARD FORM OF A LINEAR EQUATION Ax + By = C - Useful for identifying linear equations; finding x and y intercepts; and graphing a line using a table of values or x and y intercepts. Slope Formula m = y2 – y x2 – x1 - Used to find the slope of a line when given two ordered pairs.

y = kx (k = y/x is the constant of variation) Direct Variation y = kx (k = y/x is the constant of variation) Used to graph lines (the constant of variation is the slope of the line) Slope Intercept Form y = mx + b (m = slope, b = y-intercept) -Used to graph a line from an equation or write an equation from two ordered pairs. Point-Slope Form y – y 1 = m(x – x1) - Used to write an equation from two points.

Lesson 4-9 Slopes of Parallel and Perpendicular Lines Two different nonvertical lines are Parallel if and only if they have the same slope. All different vertical lines are parallel. Two different nonvertical lines are Perpendicular if and only if the product of their slopes is -1. Vertical lines are perpendicular to horizontal lines.

Lesson 4-10 Family of Functions – a set of functions whose graphs have basic characteristics in common (their graphs are the same basic shape). Parent Function – the most basic function in a family (for linear functions, the parent function is f(x) = x.

Translation of a Linear Function Transformation – a change in position or size of a figure Translation of a Linear Function Translation (slide) – a type of transformation that moves every point the same distance in the same directions When the y-intercept b is changed in the function f(x) = mx + b the graph is translated vertically.

Rotation of a Linear Function Rotation (turn) – a transformation about a point (the y-intercepts are the same, but the slope is different). When the slope m is changed in the function f(x) = mx + b it causes a rotation of the graph about the point (0,b), which changes the line’s steepness.

Reflection of a Linear Function Reflection (flip) – a transformation across a line that produces a mirror image. When the slope m is multiplied by -1 in f(x) = mx + b, the graph is reflected across the y-axis.