Function Characteristics
Consider the quadratic expression below.
-3x^2 + 5x + 12
Would this expression result in a parabola that is concave up or concave down?
Concave Down
A new car that sells for $18,000 depreciates 25% each year. Write a function that models the value of the car. Find the value of the car after 4 yr.
Is this a growth or decay problem? What is the rate of growth or decay? What is a? What is b? What is x?
a = 18,000
b = 0.75
x = 4
The polynomial identity (a-b)^2 can be expanded into which of the following expressions?
a^2 - 2ab + b^2
If given the following exponential equation:
f(x) = 250(0.88)^x
Does this equation demonstrate growth or decay? What is the rate of growth or decay?
Decay
The rate of decay is 12%
A ball is thrown into the air with an upward velocity of 36 ft/s. Its height h in feet after t seconds is given by the function h = −16t^2 + 36t + 9. In how many seconds does the ball reach its maximum height?
1.125 seconds
Angelo wants to graph the quadratic function shown.
g(x) = (x-4)(x+2)
At what points does the graph of the function intersect the x-axis?
(4, 0) and (-2, 0)
A scientist is studying bacteria and records the number of bacteria over time. The scientist determines that the function N(t) = 300(1.30)^t models the number of bacteria, N, after t hours.
What does 300 represent in this function? What does 1.30 represent in this function? What is the rate of growth or decay?
300 is the amount of bacteria the scientist started with.
1.30 is the growth factor.
The rate of growth is 30%
Consider this sequence of numbers.
6, 12, 24, 48, 96, ...
Which function represents this sequence?
f(n) = 3(2)^n
Below is a quadratic expression in factored form.
(8p − 2)(6p + 2)
What is this expression in standard form?
48p^2 + 4p − 4
A profit function for a product made by a company is shown below. In the function, x is the number of units sold and P(x) is the total profit.
P(x) = -2x^2 + 60x -250
For which values of x is the profit equal to 0?
5 and 25
The deer population in a national park is experiencing exponential growth. The current population of deer in the area is 2,500. The deer are growing at a rate of 3%. What equation can be used to predict the population of deer after 4 years?
f(x) = 2,500(1.03)^4
or
f(x) = 2,500(1 + 0.03)^4
For a physics experiment, students throw a ball into the air from the top of a school building. They write the expression -16t^2 + 48t +56 to represent the height, in feet, of the ball t seconds after it is thrown.
Based on this expression, what is the height, in feet, of the school building?
56 feet
A certain painting was purchased for $15,000. Its value is predicted to decay exponentially, decreasing by 15% each year. Which equation can be used to predict t, the number of years it would take for the painting to have a value of $10,000?
15,000(0.85)^t = 10,000
If Laura uses factoring to find the zeros of the quadratic function f(x) = 3x^2 + 10x - 8 , which of the following is the factored form of the function?
f(x) = (3x - 2)(x + 4)
A new truck that sells for $29,000 depreciates 12% each year. Write a function that models the value of the truck. Find the value of the truck after 7 years.
$11,851.59
Mitchell knows that f(x) = x^2 is the parent graph of a quadratic. What steps should he take if he wants to transform the parent graph to obtain the graph of f(x) = (x + 6)^2 - 3?
He should shift it 6 units to the left and 3 units down.
Consider the quadratic expression written in vertex form below:
y = 2(x - 4)^2 + 8
What is the location of the vertex?
(4, 8)
The width of Gabby's backyard is 4 meters shorter than its length, and its area is equal to 120 square meters. Which equation can be used to find the possible values for the length of Gabby's backyard, x?
x^2 - 4x - 120
or
x^2 - 4x = 120
The population of an endangered bird is decreasing at a rate of 0.75% per year. There are currently about 200,000 of these birds. Is the population growing or decaying? What is the growth or decay factor?
Decaying
The decay factor is 0.9925
We placed $1000 in a bank account with an annual compound interest rate of 8%. How long will it take until we have $5000 in the account?
20.9 years