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Here are a few recommended readings before getting started with this lesson.
The parabolas are shown in the graph below if there is a problem loading the applet.
When given a function, it is not always possible to use any value as an input. Sometimes the input will not make sense in the given context of the function. It could also be that the function is not defined for such value.
Function | Analysis | Domain |
---|---|---|
f(x)=3x | Multiplication by 3 is defined for all real numbers. | All real numbers |
g(x)=x | Square roots are not defined for negative numbers. | All non-negative numbers — that is, x≥0 |
h(x)=x1 | Division by zero is undefined. | All real numbers except 0 — that is, x=0 |
Given a function and its domain it is possible to study the set of all possible outputs.
The range of a function is the set of all y-values, or outputs, of the function. The range of a function depends on both the domain and the function itself. For example, consider the following functions and their defined domains.
Function | Domain |
---|---|
f(x)=2x | All integers |
g(x)=x2 | All real numbers |
h(x)=4 | All real numbers |
By analyzing the definition of each function along with the given domains, the ranges can be determined.
Function | Domain | Analysis | Range |
---|---|---|---|
f(x)=2x | All integers | The function takes any integer input and produces an output that is an even number, as each input is multiplied by 2. | All even numbers |
g(x)=x2 | All real numbers | The function takes any real number input and produces an output that is a non-negative number, as each input is squared. | All non-negative numbers — that is, y≥0 |
h(x)=4 | All real numbers | The function takes any real number input and sends it to 4. | Only the number 4 — that is, the range is {4} |
One of the characteristics of the graph of a quadratic function — a parabola — is the vertex.
a-value | Range |
---|---|
a>0 | [k,∞) |
a<0 | (-∞,k] |
P=60
LHS−2ℓ=RHS−2ℓ
LHS/2=RHS/2
Write as a difference of fractions
Calculate quotient
Rearrange equation
Commutative Property of Addition
Factor out -1
a=a+152−152
a2−2ab+b2=(a−b)2
Distribute -1
Calculate power
Since the vertex is at (15,225), the maximum area of the fence is 225, and this area is obtained by making a fence 15 feet long. It is not a coincidence that this is a square fence! Not every seedling can be protected from the deer, but Dominka and Emily did great.
It has been stated previously that the vertex of a parabola is either the absolute maximum or absolute minimum of the corresponding quadratic function. There is also an important characteristic of the quadratic function's graph that changes at this point.
A function is said to be increasing when, as the x-values increase, the values of f(x) also increase. On the other hand, the function is considered decreasing when, as x increases, f(x) decreases. An increasing interval is an interval of the independent variable where the function is increasing. A decreasing interval is an interval of the independent variable when the function is decreasing.
The two of them look around for the answer. They find a signboard that explains that the cart uses software that controls the lights. By knowing the function and stating the increasing and decreasing intervals, the lights can switch accordingly. The signboard shows the blueprints.
The blueprints also show the quadratic function that describes the parabolic section of the roller coaster.A quadratic function might cross the x- and y-axes. These points are known as intercepts.
The x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis. The y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis. The y-intercept of an equation is also known as its initial value.
When talking about functions, the x-intercepts are the zeros of the function. Sometimes, only one coordinate of these points is referenced. For example, if the x-intercept lies at (a,0), it can be said that the x-intercept is at x=a. The same is true for the y-intercept. A relation can have several intercepts. A function can have multiple x-intercepts, but it can only have one y-intercept.Dominika and Emily, enjoying the theme park, want to try Shoot 'N' Hoop. If they make their one shot attempt, they will win a teddy bear. As the ball goes up, Dominika imagines her shot being modeled by a quadratic function.
The function representing Dominika's shot is graphed on the following coordinate plane. Take note that the drawn path of the ball in the air in the applet versus when mapped on the graph looks slightly different.
The height of the ball is represented by y, and time the ball is in the air after Dominika shoots the ball is x. The worker hosting the game tells Dominika and Emily that if they can answer the following math questions correctly, he will give them a second teddy bear.
Substitute values
Calculate power
-a(-b)=a⋅b
Multiply
Add terms
Put minus sign in numerator
Distribute -1
Calculate root
Parts of the graph of a quadratic function can be either above or below the x-axis. The intervals of x-values where this happens receive a special name.
A function is said to be positive where its graph is above the x-axis and is said to be negative where its graph is below the x-axis.
A positive interval is an interval for which the function is positive. Likewise, a negative interval is an interval for which the function is negative. The graph above shows two negative intervals and two positive intervals. Each interval is described in terms of the x-values.Emily and Dominika are relaxing on a bench at the theme park. Next to them, a bluebird appears in a bird bath! The bath seems to be a bit low on water, however. They look up the blueprints of similar birdbaths online and find a cross section of one that includes its quadratic function.
That birdbath's manual suggests to fill the bath with water up to the x-axis. This way, the birdbath will be filled in its negative interval. Knowing that the x-intercepts are at x=0.5 and x=5.5, state the the positive and negative intervals.Positive Intervals: [0,0.5) and (5.5,6]
Negative Interval: (0.5,5.5)
The positive and negative intervals do not include the points where the function is equal to zero.
In general, the domain of quadratic functions consists of all real numbers. Investigating what happens to the function as the x-values increase or decrease infinitely is valuable in better understanding quadratic functions.
up.
down.
down and upor as
down-up.
Since quadratic functions are polynomial functions of degree 2 their end behavior is either up and up or down and down, all depending on the sign of the leading coefficient.