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Standard Form Section 5-5 Goals Goal • To graph linear equations using intercepts. • To write linear equations in standard form. Rubric 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. Ax By C •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 xaxis 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. 5x – 2y = 10 5x – 2(0) = 10 5x – 0 = 10 5x = 10 x=2 The x-intercept is 2. To find the y-intercept, replace x with 0 and solve for y. 5x – 2y = 10 5(0) – 2y = 10 0 – 2y = 10 –2y = 10 y = –5 The y-intercept is –5. Your Turn: Find the x- and y-intercepts. The graph intersects the yaxis at (0, 3). The y-intercept is 3. The graph intersects the xaxis at (–2, 0). The x-intercept is –2. Your Turn: Find the x- and y-intercepts. –3x + 5y = 30 To find the x-intercept, replace y with 0 and solve for x. –3x + 5y = 30 –3x + 5(0) = 30 –3x – 0 = 30 –3x = 30 x = –10 The x-intercept is –10. To find the y-intercept, replace x with 0 and solve for y. –3x + 5y = 30 –3(0) + 5y = 30 0 + 5y = 30 5y = 30 y=6 The y-intercept is 6. Your Turn: Find the x- and y-intercepts. 4x + 2y = 16 To find the x-intercept, replace y with 0 and solve for x. 4x + 2y = 16 4x + 2(0) = 16 4x + 0 = 16 4x = 16 x=4 The x-intercept is 4. To find the y-intercept, replace x with 0 and solve for y. 4x + 2y = 16 4(0) + 2y = 16 0 + 2y = 16 2y = 16 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. x-intercept: 3x – 7y = 21 3x – 7(0) = 21 3x = 21 x=7 y-intercept: 3x – 7y = 21 3(0) – 7y = 21 –7y = 21 y = –3 Example: Continued x-intercept: x = 7 and y-intercept: y = -3 Step 2 Graph the line. Plot (7, 0) and (0, –3). x 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 x+y=4 Add x to both sides. Example: Continued x+y=4 Step 2 Find the intercepts. x-intercept: x+y=4 x+0=4 x=4 y-intercept: x+y=4 0+y=4 y=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: 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 + 4y = –12 –3x + 4(0) = –12 –3x = –12 –3(0) + 4y = –12 4y = –12 x=4 y = –3 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 –x + 3y = –6 Write the equation in standard form. Your Turn: Continued –x + 3y = –6 Step 2 Find the intercepts. x-intercept: –x + 3y = –6 –x + 3(0) = –6 –x = –6 y-intercept: –x + 3y = –6 –(0) + 3y = –6 3y = –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. Y Equations of Horizontal Lines • 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. (0,b) Y-axis (x,b) X-axis X Y Horizontal Line • The equation for the horizontal line is still y = mx + b ( Slope Intercept Form ). Where m is: m= DY DX = Y-axis (b – b) (x – 0) DY = 0 =0 (0,b) (x,b) DX X-axis X Y Horizontal Line • Because the value of m is 0, y = mx + b becomes Y-axis y=b (A Constant Function) (0,b) (x,b) X-axis X Y Example 1: Horizontal Line • 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 m= DY DX = (2 – 2) (5 – 0) =0 (0,2) (5,2) DX DY = 0 X-axis X Y Example 1: Horizontal Line • Because the value of m is 0, y = 0x + 2 becomes Y-axis y=2 (A Constant Function) (0,2) (5,2) X-axis X Y Your Turn: • Find the equation for the lines passing through the following points. 1.) (3,2) & ( 8,2) y=2 2.) (-5,4) & ( 10,4) y=4 3.) (1,-2) & ( 7,-2) y = -2 4.) (4,3) & ( -2,3) y=3 X Y Equations of Vertical Lines • 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 (a,y) (a,0) X-axis X Y Vertical Line • The equation for the vertical line is x = a ( a is the X-Intercept of the line). Because m is: Y-axis (a,y) m= DY DX = (y – 0) = Undefined (a – a) (a,0) X-axis X Y Vertical Line • 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=a (The equation of a vertical line) (a,y) (a,0) X-axis X Y Example 2: Vertical Line • Let’s look at a line with no y-intercept b, an x-intercept a, passing through (3,0) and (3,7). Y-axis (3,7) (3,0) X-axis X Y Example 2: Vertical Line • The equation for the vertical line is: x = 3 ( 3 is the X-Intercept of the line). Because m is: Y-axis (3,7) m= DY DX = (7 – 0) (3 – 3) = 7 0 = Undefined (3,0) X-axis X Y Your Turn: • Find the equation for the lines passing through the following points. 1.) (3,5) & ( 3,-2) x=3 2.) (-5,1) & ( -5,-1) x = -5 3.) (1,-6) & ( 1,8) x=1 4.) (4,3) & ( 4,-4) x=4 X 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