graphing Linear equations & functions Algebra 1 LE graphing Linear equations & functions

Front Flap Notes Point–Slope Form: Slope–Intercept Form: Standard Form: Slope Equation:

Anatomy of a coordinate plane  

Functions Function: a relation that pairs each input with exactly one output. Input Output x coordinate y coordinate domain range independent dependent

graphing w/ A table Graph the following equation using a table y – 3 = 2(x + 3) Point–Slope Form   x y

graphing w/ A table Graph the following equation using a table y = 1/4x – 2 Slope–Intercept Form   x y

graphing w/ A table Graph the following equation using a table 6x + 3y = 9 Standard Form   x y

Graphing – Special Cases Horizontal and Vertical Lines

More about Functions “pairs each input with exactly one output” 5 12 6 15 8 7 18 27 x y 5 12 6 15 7 18 8 21 9 24

More about Functions The Vertical Line Test

What is slope? Given two points on a line, slope is defined as the change in y over the change in x. Also helpful to think of as rate of change. Formula to Calculate Slope is often abbreviated as m

Slope from a graph Calculate the slope of the line: P1 (–4, 2) P2 is (2, 6)Identify point #1 and #2. m = (6 – 2)/(2 – (–4)) Substitute m = 4/6 Evaluate m = 2/3 Simplify

Slope from A Graph Calculate the slope of the line passing through (0, 4) and (1, –2). P1 is (0, 4) P2 is (1, –2)Identify point #1 and #2. m = (–2 – 4)/(1 – 0) Substitute m = –6/1 Evaluate m = –6 Simplify

Parallel Lines Are these two lines parallel? Parallel Slopes are Equal Perpendicular Slopes are Opposite Reciprocals   (3, 5) (5, 2) (–4, –1) (–2, –4)

classification of slopes

classification of slopes

graphing w/ intercepts What are Intercepts? Intercepts are the points of intersection between a graph of a linear equation and the x or y axis. Does every linear equation have intercepts? Yes, every linear equation will have either one or two intercepts depending on the slope of the line.

graphing w/ intercepts Graph using the x and y intercepts 3x + 7y = 21 To find x-intercept, substitute 0 for y and solve for x. To find y-intercept, substitute 0 for x and solve for y.  

Practice Find the intercepts of each equation: 3x – 3y = 6 x-intercept = 2, y-intercept = –2 4x – 2y = 10 x-intercept = 5/2, y-intercept = –5 –3x + 5y = –15 x-intercept = 5, y-intercept = –3

#3 – Problem Solving Helldog is planning on scoring 36 points in the faculty- student basketball game, scoring a combination of two- point and three-point shots. Model this scenario with a linear equation. 2x + 3y = 36 What are the intercepts and what do they mean? x-intercept: 18; y-intercept: 12. Amount of either one (without any of the other) he could make to reach his goal. Give one combination of two’s and three’s he could make to reach his goal. 3 two’s and 10 three’s.

Graph w/ Slope–Intercept Graph: y = –2x – 5 Graph will go through point (0, –5) and have a slope of –2.  

Graph w/ Slope–Intercept Graph: y = 3/2x + 2 Graph will go through point (0, 2) and have a slope of 3/2.  

Graph w/ Point–Slope Graph: y – 2 = 3(x + 3) Graph will go through point (–3, 2) and have a slope of 3.  

Graph w/ Point–Slope Graph: y + 4 = 2/3(x – 1) Graph will go through point (1, –4) and have a slope of 2/3.  

#1d – graphing w/ A table Graph the following equation using a table f(x) = 3x + 1 Function Notation   x f(x)

#8 & 9 – Function notation f(x) = mx + b f(x) is much like y The function in this example is named f, however any letter can be used to name a function Read as “f of x” or “the value of f at x” DOES NOT MEAN f multiplied by x

#8 – Evaluate a Function Evaluate f(x) = 4x + 7, for x = –4, 3, 7 f(–4) = –9 Ordered Pair (–4, –9) f(3) = 4(3) + 7 f(3) = 19 Ordered Pair (3, 19) f(7) = 4(7) + 7 f(7) = 35 Ordered Pair (7, 35)

#9 – Solve a Function Solve f(x) = –3x – 6, so f(x) = 18. Trying to find x value to make statement true. 18 = –3x – 6 Rewrite w/ f(x) as 18 24 = –3x –8 = x Solve equation for x f(–8) = 18 x must be equal to –8

#3 – Practice (5, 2) & (4, –1) (–2, 3) & (4, 6) (9/2, 5) & (1/2, –3) m = (–1 – 2)/(4 – 5) m = 3 (–2, 3) & (4, 6) m = (6 – 3)/(4 – (–2)) m = 1/2 (9/2, 5) & (1/2, –3) m = (–3 – 5)/(1/2 – 9/2) m = 2