Slope-Intercept Form
What do the letters (m) and (b) represent in the equation (y = mx + b)?
m: slope or rate of change
y: y-intercept
Define the slope of a line.
The slope represents the rate of change of an equation. It is equal to rise/run or the change in y over the change in x.
Explain the significance of intercepts on a linear graph.
Bonus points: How many x and y intercepts does a linear graph have?
The x and y intercepts represent where the line of the equation crosses both the x and y axes. This highlights key points on a graph that can be used for plotting and interpreting graphs.
A maximum of one x intercept and one y intercept but it could depend on the domain and range or the real life application of the equation.
Write the general form of a linear equation.
Ax+Bx+C=0
Define the domain and range of a function.
The domain is the set of all possible input values (x-values) for the function.
The range is the set of all possible output values (y-values) for the function.
For the equation y=-4x+5, determine the value of y when x = 23
y = -87
If a line has a slope of (-3), how does it behave on a graph?
It trends downwards, every point is down three units and over to the right one unit
What is the coordinate of the y-intercept of the line (y = 4x + 2)?
(0,2)
What is the value of (A) in the equation (2x - 4y + 8 = 0)?
A = 2
What is the range of the function (f(x) = 2x + 3)?
y is an element of all the real numbers
For the equation y=(1/2)x-6, determine the value of x when y = 57
x = 126
What is the slope of a vertical line?
UND, can't ever never ever never divide by 0.
What is the coordinate of the x-intercept for the equation (y = -5x + 50)?
(10,0)
Convert (y = (3/4)x+2) into general form.
0=3x-4y+8
What is the domain of the equation y=2x-5
x is an element of all the real numbers
Convert the equation in general form, 2x−3y=6, to slope-intercept form (y=mx+b)
y = (2/3)x-2
Calculate the slope between the points (2, 3) and (5, 11).
Bonus points: can you construct the formula in the form y=mx+b for this line?
slope = 8/3
y=(8/3)x-(7/3)
What are the coordinates of the x and y intercept of the equation (3x + 2y = 6)?
x-intercept: (2,0)
y-intercept: (0,3)
How can you identify if a line is vertical or horizontal in general form?
A vertical line will have no y term, while a horizontal line will have no x term.
What is the domain and range for the equation y=7
D: x is an element of the real numbers
R: y=7
Keely is choreographing a new dance routine where she incorporates a moving platform. The platform starts at a height of 5 feet and rises at a constant rate of 0.75 feet per second. Construct a linear equation in slope-intercept form (y=mx+b) that represents the height y of the platform in feet after x seconds. Then, use your equation to determine:
a) The height of the platform after 12 seconds.
b) How long it will take for the platform to reach a height of 20 feet.
y=(3/4)x+5
a) 14 ft
b) 20 seconds
Cash is training his horse for barrel racing. He tracks the horse's speed during practice runs. On the first day, Cash times the run. In 5 seconds the horse has run 40 ft and by 12 seconds the horse has run 96 ft. If Cash plots these two points on a graph with time on the x-axis and distance on the y-axis, what is the slope of the line connecting these points? What does this slope represent in terms of the horse's performance?
slope = 8 ft/s
For every second that passes, the horse covers 8 ft of ground. The horse is running at a constant pace from start until finish. Is this realistic?
At Liam's martial arts dojo, there's a linear relationship between the number of classes attended and the belt level achieved. When a student has attended 0 classes, they're at belt level 1 (white belt). After 20 classes, they reach belt level 5. The maximum belt level is 20. What is the equation for this relationship? What are the x and y intercepts and what do they represent?
Bonus points: what is the domain and range?
y=(1/4)x+1
y-intercept: (0,1)--> this represents the starting rank before you attend any classes.
Range: 1<(or equal to) y<(or equal to) 20
x-intercept: (-4,0) --> this doesn't make sense in the context of the question. You can't attend -4 classes. So the domain is restricted to 0<(or equal to) x<(or equal to) 76
Zak, a devoted Saints fan, is analyzing the team's scoring pattern over the past few games. He notices that the relationship between the number of games played (x) and the total points scored (y) can be represented by a linear equation in general form: 3x−2y+12=0. If the Saints have played 8 games so far this season, how many total points have they scored? What is this equation in slope-intercept form?
18 games
y=(3/2)x+6
Connor is analyzing the brake pad wear on a fleet of delivery trucks. He discovers that the thickness of the brake pads decreases linearly with the number of miles driven. A new set of brake pads is 20 mm thick, and they wear down by 0.05 mm every 1000 miles driven. The brake pads need to be replaced when they reach a thickness of 4 mm. Create a linear function to model this relationship, and determine the domain and range of this function.
y=(-5/100000)x+20
or
y=(-0.05/1000)x+20
D: 0<(or equal to) x <(or equal to) 320000
R: 4<(or equal to) y <(or equal to) 20
Does this make sense?