Domain and Range of Quadratic Functions

End Behavior of a Graph End Behavior of a graph – Given a quadratic function in the form 𝒇 𝒙 =𝒂 𝒙 𝟐 + bx + c or 𝒇 𝒙 =𝒂 𝒙 −𝒉 𝟐 the quadratic function is said to open up if a > 0 and open down if a < 0.

If a > 0 then f has a minimum at the x -coordinate of the vertex, f is decreasing for x-values less than (or to the left of) the vertex, and f is increasing for -values greater than (or to the right of) the vertex. If a < 0 then has a maximum at x-coordinate of the vertex f is increasing for x -values less than (or to the left of) the vertex, f is decreasing for x-values greater than (or to the right of) the vertex.

The minimum value of a function is the least possible y-value for that function. The maximum value of a function is the greatest possible y-value for that function. The highest or lowest point on a parabola is the vertex. Therefore, the minimum or maximum value of a quadratic function occurs at the vertex.

Review Domain is all the possible x values of a function Range is all the possibly y values of a function Unless a specific domain is given, the domain of a quadratic function is all real numbers. One way to find the range of a quadratic function is by looking at its graph.

What patterns do we see? When we are trying to figure out the domain of any function the question we should ask ourselves is: What possible values could this function take on for x? We can ask the same question for range. What possible values could this function take on for y?

What patterns do we see? Sometimes people get confused and state domain and range in terms of what a function cannot be. THIS IS WRONG!! Always state the domain and range in terms of what can be!!

What patterns do we see? Unless a parabola has dots at its end or we are specifically told that it does not continue to extend indefinitely, we can make the presumtion that it will always have the same domain: All real numbers −∞,∞

Why is this? Think about it: As we move up from the vertex of (0,0) we notice that the parabola continues to get wider and wider. This pattern will continue forever. So there will be no restriction on what x can possibly be.

What about the range? The range will change from graph to graph We can see from the previous graph that it will never go below the y value of “0”. Therefore, y can only be greater than 0. It will still belong to real numbers because there is an unbroken line connecting all the points. R: [0, ∞, ) R: y> 0

Another Example Find the domain and range of the following:

Real World Applications Imagine the height of a ball thrown off a building is modelled by the equation Where t is time in seconds and h is height in meters What would be an appropriate domain and range?

Real World Applications We can see that the x intercepts are approximately –3.5 and +7.5. However, it is not realistic to have negative time therefore we would modify the domain to:

Real World Applications Similarly, it wouldn’t make sense for the ball to go below ground so the range would be as follows:

Pop Quiz Find the Domain and Range of the following relations.

Graph 1 Graph 2

Graph 3 Graph 4

Answers: 1. 2. 3. 4. 5. Graph 5

So in conclusion: Don’t just apply a blanket idea to everything…..look at the circumstances as well.

Lesson Quiz Use the graph for Problems 3-5. 1. Identify the vertex. 2. Does the function have a minimum or maximum? What is it? 3. Find the domain and range. (5, –4) maximum; –4 D: all real numbers; R: y ≤ –4