Question

In Exercises 49 and \(50,\) graph a function on the interval [-2,5] having the given characteristics. Absolute maximum at \(x=-2\) Absolute minimum at \(x=1\) Relative maximum at \(x=3\)

Short answer

Since exact values are not given, the graph will depend on arbitrary chosen y-values for the points of maxima and minima, but will properly comply with the given conditions. It starts at an absolute maxima at \(x=-2\), decreases to reach an absolute minimum at \(x=1\), then rises to a relative max at \(x=3\), and proceeds from there without exceeding previous maxima or going below the absolute minimum within the interval [-2,5].

Step-by-step solution

Understand the Definitions of Given Terms

The absolute maximum point on a graph is where the function achieves its highest value in the entire domain, in this case, the interval [-2,5]. Conversely, the absolute minimum is where it hits its lowest value. A relative maximum is a point where the function achieves a high value, but only in comparison to the points directly around it. There could be other points higher up elsewhere on the graph.

Plot the Given Points

Knowing that the function has an absolute maximum at \(x=-2\), an absolute minimum at \(x=1\), and a relative maximum at \(x=3\), plot these three points. As there is no information given on the exact y-values of these points, you can choose arbitrary y-values that satisfy the conditions of maxima and minima.

Connect the Dots

Now that the main points are in place, the next step is to draw the function line. Here you need to ensure that the function has the correct shape and satisfies all the given conditions. The line starting from the point (\(-2, y_{1}\)) where \(y_{1}\) is an arbitrary y-value representing absolute maximum should proceed downwards to reach the point (\(1, y_{2}\)) where \(y_{2}\) would be a much lower value than \(y_{1}\) as it represents the absolute minimum. Then, from this point, the line should go upwards reaching the relative maximum at point (\(3, y_{3}\)) where \(y_{3}\) is less than \(y_{1}\) but more than \(y_{2}\). Lastly, the function will proceed either upward or downward after \(x=3\) but will not exceed \(y_{1}\) or go below \(y_{2}\) on the interval [-2,5].

Key concepts

Absolute Maximum

When you're graphing a function, an absolute maximum is a feature you'll definitely want to spot. It's a point on the graph where the function's value is higher than at any other point in the function's domain. Think of it like the peak of a mountain; it's the highest point in the entire mountain range. In the exercise provided, the function has its absolute maximum at the beginning of the designated interval, when x=-2. It's crucial to remember that this maximum isn't just a local observation—it's the champion of heights throughout the entire domain you're considering.

For those dealing with this concept on homework or exams, here's a pro tip: when you’re asked to graph a function with an absolute maximum (or minimum), start by placing this highest (or lowest) point first before plotting the rest of the graph. This will give you a clear boundary to work within when drawing the function's curve.

Absolute Minimum

The sister concept to an absolute maximum is an absolute minimum. This feature on a graph marks the location where the function takes the lowest value across its entire domain. Imagine you're looking at a valley—the absolute minimum is the lowest point, right at the bottom. In the given exercise, the function bottoms out at an x-value of 1, creating its absolute minimum. It's a full valley floor in the range we care about, which is the interval from -2 to 5.

If you're sketching a graph and you need to show the absolute minimum, make sure that no other point dips below the y-value you select for this minimum. This lowest point is a critical anchor for the rest of your graph. Keeping the 'valley floor' idea in mind can help ensure you place this minimum correctly and avoid any potential holes or unwanted dips in other sections of your graph.

Relative Maximum

Now, let's climb a little hill with the concept of a relative maximum. It's like a small bump or hilltop on your graph that's not necessarily the ultimate peak, but it's higher than its immediate surroundings. If you're hiking along a trail, a relative maximum is a point where you'd stop to catch your breath because you've just walked up a small incline even if it's not the highest peak on your route. In the exercise scenario, our path hits a relative high at x=3. It's not the peak of the whole function, but it sticks out because it's a point where the function climbs higher before dropping off again or leveling out relative to the nearby points on the graph.

Understanding this can be a game changer when graphing: a relative maximum doesn't have to be the highest point overall, just higher than its neighbors. This knowledge helps in accurately drawing intermediate highs and lows within the graph, ensuring that each relative maximum is correctly represented with respect to its local area of the function.

Function Domain

When you're plotting the course of a function, you need to know the lay of the land—the function domain. It sets the stage for where the function lives and plays. Technically speaking, the domain of a function encompasses all the x-values (or inputs) that the function will accept. It's like the 'address range' for your graph's x-values. In the context of our exercise, the function's domain is the interval [-2, 5]. This means we're only interested in what happens to the function between these x-values, including them both. Anything happening outside of this interval? That's like the dark side of the moon for our current mission—it's there, but it doesn't concern us right now.

Understanding the domain is incredibly helpful—it tells you where to start and stop your graph. Whether you are dealing with visuals or calculations, remembering that the domain is basically the function's 'home turf' helps keep your work relevant and focused on the right segment of the x-axis.