Equations of a straight line
What does 'm' stand for in the following equations of a straight line:
y-y1 = m(x-x1) and y = mx + c
gradient (SLOPE IS WRONG AND NOT ACCEPTED AS AN ANSWER)
Determine the image of the following quadratic function at x = 5
f(x) = 3x2 - 1.5x + 7
To find the image, you substitute the value of x into the function, so f(5) = 3(5)2 - 1.5(5) + 7 = 74.5
The domain of a function is associated with the ___ values and the range of a function is associated with the ____ values. (Both blanks must be correct to earn the points)
domain is associated with x values (inputs) and range is associated with y-values (outputs)
Name one parent function that has a domain of
[x: -∞ < x < ∞]
linear, quadratic, cubic, exponential growth/decay, trigonometric
Determine the gradient between the following two points: (-4,3) and (-7,5)
Use the gradient formula: (y2-y1)/(x2-x1)
(5 - 3)/(-7 - (-4)) = 2/(-7 + 4) = -2/3
In order for a function to have an inverse, that function must have what characteristic?
be one-to-one, every input has only one output and every output as only one input
What are the zeros of a function?
True or false: The value of an asymptote is included in the domain/range.
False
Parallel lines have the _____________ gradient while perpendicular lines have gradients that are the _____________________, _____________________ of each other. You must fill in all blanks to earn the points for this question.
same, opposite, reciprocal (or same, reciprocal, opposite)
Describe one way you can find the inverse of a function?
switch x and y in the equation and use inverse operations isolate y, reflect the function across the line y=x, switch the x and y values of some of the points of the function, graph those points, and determine the line that goes through them
The minimum or maximum turning point of the graph.
Ms. Reinbold was heating up water for tea and asked me to determine the function that models the rate at which the water heats up.
I determined this function to be: T(t) = 0.1e0.03t + 20 where t is the time in seconds and T(t) is the temperature of the water in degrees celsius.
What is the temperature of the water when it is placed in the kettle?
Plug in 0 for t to get 20.1 degrees Celsius
The equation for the line that represents Broad Street is given as: y = -2x + 5. Find the equation for Malvern Avenue, which is perpendicular to Broad Street, if you are standing on Malvern Avenue at point (-3,2).
the gradient of Malvern Avenue is the opposite reciprocal of the gradient of Broad Street, so 1/2
substitute the gradient (1/2) and the point (-3,2) into point-gradient form and then simplify the double negative
y - 2 = 1/2(x + 3)
Determine the inverse of this function:
f(x) = 5x - 3
y = 5x - 3 (replace f(x) with y)
x = 5y - 3 (switch x and y)
x + 3 = 5y
x/5 + 3/5 = y (isolate y)
f-1(x) = x/5 + 3/5 (replace y with f-1(x))
What do we call the vertical or horizontal line that the graph of a function approaches but does not touch?
An asymptote
Ms. Reinbold was heating up water for tea and asked me to determine the function that models the rate at which the water heats up.
I determined this function to be: T(t) = 0.1e0.03t + 20 where t is the time in seconds and T(t) is the temperature of the water in degrees Celsius.
Determine the horizontal asymptote of this graph.
y = 20, graph the function and trace to the left to observe the y-value that the graph is approaching but never touches
You are driving from your house to school. You note the time you leave your house to be 8:15 and the time you arrived at school to be 8:30. You also know that you live 5 miles from the school. You want to determine the equation that models your trip to school, so you create the following two points:(0,0) and (15,5) where x represents time in minutes and y represents the distance traveled in miles. Determine the gradient between these two points and interpret it's meaning in the context of the problem.
Use the gradient formula: (y2-y1)/(x2-x1)
(5-0)/(15-0) = 5/15 = 1/3 miles per minute
This represents the average speed that you traveled to school at in miles per minute
Find the inverse of this function:
f(x) = 3x3
y = 3x3 (replace f(x) with y)
x = 3y3 (switch x and y)
1/3x = y3
∛(1/3x) = y (isolate y)
f-1(x) = ∛(1/3x) (replace y with f-1(x))
Name one parent function that has an asymptote.
Exponential growth, exponential decay, rational (inverse)
Ms. Reinbold was heating up water for tea and asked me to determine the function that models the rate at which the water heats up.
I determined this function to be: T(t) = 0.1e0.03t + 20 where t is the time in seconds and T(t) is the temperature of the water in degrees Celsius.
Determine how long it takes for the water to boil. Water boils at 100 degrees Celsius
Since temperature is the 'y' value, you can use numeric solver to solve for 'x' or the time in seconds, t. Press math and scroll down to find Numeric Solver and press enter. Then type 0.1e0.03x + 20 into E1 and 100 into E2, press enter, then press alpha, enter and it will solve for x. (If you do not press alpha enter, it will not solve). X = 22.8 or 223 seconds.