Standard Form Section 5-5

Goals Goal Rubric To graph linear equations using intercepts. To write linear equations in standard form. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary x-intercept Standard form of a linear equation

Standard Form Another way to determine whether a function is linear is to look at its equation. A function is linear if it is described by a linear equation. A linear equation is any equation that can be written in the standard form shown below.

Standard Form Notice that when a linear equation is written in standard form. x and y are both on the same side of the equal sign. 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.

Axial Intercepts A y-intercept is the y-coordinate of any point where a graph intersects the y-axis. The x-coordinate of this point is always 0. An x-intercept is the x-coordinate of any point where a graph intersects the x-axis. The y-coordinate of this point is always 0.

Example: Finding Intercepts From a Graph Find the x- and y-intercepts. The graph intersects the y-axis at (0, 1). The y-intercept is 1. The graph intersects the x-axis at (–2, 0). The x-intercept is –2.

Example: Intercepts From Equation Find the x- and y-intercepts. 5x – 2y = 10 To find the x-intercept, replace y with 0 and solve for x. To find the y-intercept, replace x with 0 and solve for y. 5x – 2y = 10 5x – 2(0) = 10 5x – 2y = 10 5(0) – 2y = 10 5x – 0 = 10 5x = 10 0 – 2y = 10 –2y = 10 x = 2 The x-intercept is 2. y = –5 The y-intercept is –5.

Your Turn: Find the x- and y-intercepts. The graph intersects the y-axis at (0, 3). The y-intercept is 3. The graph intersects the x-axis at (–2, 0). The x-intercept is –2.

Your Turn: Find the x- and y-intercepts. –3x + 5y = 30 –3x + 5y = 30 To find the y-intercept, replace x with 0 and solve for y. To find the x-intercept, replace y with 0 and solve for x. –3x + 5y = 30 –3(0) + 5y = 30 –3x + 5y = 30 –3x + 5(0) = 30 0 + 5y = 30 5y = 30 –3x – 0 = 30 –3x = 30 x = –10 The x-intercept is –10. y = 6 The y-intercept is 6.

Your Turn: Find the x- and y-intercepts. 4x + 2y = 16 To find the y-intercept, replace x with 0 and solve for y. To find the x-intercept, replace y with 0 and solve for x. 4x + 2y = 16 4(0) + 2y = 16 4x + 2y = 16 4x + 2(0) = 16 0 + 2y = 16 2y = 16 4x + 0 = 16 4x = 16 x = 4 The x-intercept is 4. y = 8 The y-intercept is 8.

Graphing Using Standard Form For any two points, there is exactly one line that contains them both. This means you need only two ordered pairs to graph a line. It is often simplest to find the ordered pairs that contain the x and y intercepts. The x and y intercepts can easily be found from standard form. Therefore, to graph a linear equation in standard form, use the x and y intercepts.

Example: Graphing Standard Form Use intercepts to graph the line given by the equation. 3x – 7y = 21 Step 1 Find the intercepts. y-intercept: x-intercept: 3x – 7y = 21 3x – 7(0) = 21 3x – 7y = 21 3(0) – 7y = 21 3x = 21 –7y = 21 x = 7 y = –3

Example: Continued x-intercept: x = 7 and y-intercept: y = -3 Step 2 Graph the line. x Plot (7, 0) and (0, –3). Connect with a straight line.

Example: Graphing Standard Form Use intercepts to graph the line given by the equation. y = –x + 4 Step 1 Write the equation in standard form. y = –x + 4 +x +x Add x to both sides. x + y = 4

Example: Continued y-intercept: x-intercept: x + y = 4 Step 2 Find the intercepts. y-intercept: x-intercept: x + y = 4 0 + y = 4 x + y = 4 x + 0 = 4 y = 4 x = 4

Example: Continued x-intercept: x = 4 and y-intercept: y = 4 Step 3 Graph the line. Plot (4, 0) and (0, 4). Connect with a straight line.

Your Turn: x-intercept: y-intercept: Use intercepts to graph the line given by the equation. –3x + 4y = –12 Step 1 Find the intercepts. x-intercept: y-intercept: –3x + 4y = –12 –3x + 4(0) = –12 –3x = –12 –3x + 4y = –12 –3(0) + 4y = –12 4y = –12 y = –3 x = 4

Your Turn: Continued x-intercept: x = 4 and y-intercept: y = -3 Step 2 Graph the line. Plot (4, 0) and (0, –3). Connect with a straight line.

Your Turn: Use intercepts to graph the line given by the equation. Step 1 Write the equation in standard form. Multiply both sides by 3, the LCD of the fractions, to clear the fraction. 3y = x – 6 Write the equation in standard form. –x + 3y = –6

Your Turn: Continued y-intercept: x-intercept: –x + 3y = –6 Step 2 Find the intercepts. y-intercept: x-intercept: –x + 3y = –6 –x + 3y = –6 –(0) + 3y = –6 –x + 3(0) = –6 3y = –6 –x = –6 x = 6 y = –2

Your Turn: Continued x-intercept: x = 6 and y-intercept: y = -2 Step 3 Graph the line. Plot (6, 0) and (0, –2). Connect with a straight line.

Equations of Horizontal and Vertical Lines

Equations of Horizontal and Vertical Lines Equation of a Horizontal Line A horizontal line is given by an equation of the form y = b where b is the y-intercept. Note: m = 0. Equation of a Vertical Line A vertical line is given by an equation of the form x = a where a is the x-intercept. Note: m is undefined.

Equations of Horizontal Lines Y X Let’s look at a line with a y-intercept of b, a slope m = 0, and let (x,b) be any point on the Horizontal line. Y-axis X-axis (0,b) (x,b)

Horizontal Line Where m is: Y X The equation for the horizontal line is still y = mx + b ( Slope Intercept Form ). Where m is: Y-axis X-axis = (b – b) DY m = DY = 0 DX (0,b) (x,b) = 0 DX (x – 0)

Horizontal Line y = b (A Constant Function) X Because the value of m is 0, y = mx + b becomes Y-axis X-axis y = b (A Constant Function) (0,b) (x,b)

Example 1: Horizontal Line Y X Let’s find the equation for the line passing through the points (0,2) and (5,2) y = mx + b ( Slope Intercept Form ). Where m is: Y-axis X-axis = (2 – 2) DY m = = 0 DX (5 – 0) DY = 0 DX (0,2) (5,2)

Example 1: Horizontal Line Y X Because the value of m is 0, y = 0x + 2 becomes Y-axis X-axis y = 2 (A Constant Function) (0,2) (5,2)

Your Turn: Y X Find the equation for the lines passing through the following points. 1.) (3,2) & ( 8,2) 2.) (-5,4) & ( 10,4) 3.) (1,-2) & ( 7,-2) 4.) (4,3) & ( -2,3) y = 2 y = 4 y = -2 y = 3

Equations of Vertical Lines Y X Let’s look at a line with no y-intercept b, an x-intercept a, an undefined slope m, and let (a,y) be any point on the vertical line. Y-axis X-axis (a,0) (a,y)

Vertical Line The equation for the vertical line is Because m is: Y X x = a ( a is the X-Intercept of the line). Because m is: Y-axis X-axis (a,0) (a,y) = (y – 0) DY m = = Undefined DX (a – a)

Vertical Line x = a (The equation of a vertical line) Y Because the value of m is undefined, caused by the division by zero, there is no slope m. x = a becomes the equation Y-axis X-axis x = a (The equation of a vertical line) (a,0) (a,y)

Example 2: Vertical Line Y X Let’s look at a line with no y-intercept b, an x-intercept a, passing through (3,0) and (3,7). Y-axis X-axis (3,0) (3,7)

Example 2: Vertical Line Y X The equation for the vertical line is: x = 3 ( 3 is the X-Intercept of the line). Because m is: Y-axis X-axis (3,0) (3,7) DY = (7 – 0) = 7 m = = Undefined DX (3 – 3)

Your Turn: Y X Find the equation for the lines passing through the following points. 1.) (3,5) & ( 3,-2) 2.) (-5,1) & ( -5,-1) 3.) (1,-6) & ( 1,8) 4.) (4,3) & ( 4,-4) x = 3 x = -5 x = 1 x = 4

Linear Equations

Joke Time Why don’t blind people go skydiving? Because it scares the bejesus out of the dogs! What kind of horses go out after dark? Nightmares! Why did the skeleton go to the party alone? He had no body to go with him.

Assignment 5-5 Exercises Pg. 354 - 356: #8 – 72 even