Hey guys! Let's dive into a super interesting math problem today. We're going to break down a function that Gerald graphed and figure out some key things about it. Get ready, because we're about to make math super clear and fun!
Understanding the Function f(x)=(x-3)^2-1
Okay, so Gerald graphed the function f(x) = (x - 3)² - 1. At first glance, this might look a bit intimidating, but trust me, it's totally manageable. The key thing to recognize here is that this is a quadratic function, and more specifically, it's in what we call vertex form. Vertex form is super helpful because it tells us a lot about the graph without having to do a ton of calculations.
The general form for vertex form is f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The vertex is basically the turning point of the parabola—it’s either the lowest point (if the parabola opens upwards) or the highest point (if the parabola opens downwards). In our case, a is 1, h is 3, and k is -1. This means the vertex of Gerald's graph is at the point (3, -1). Remember that the sign in the parenthesis is opposite of what you might expect, so (x - 3) means we shift the graph 3 units to the right.
Why is this important? Well, the vertex is like the anchor of our parabola. It helps us understand the overall shape and position of the graph. Since the coefficient a is positive (1 in our case), the parabola opens upwards, meaning it has a minimum value. This minimum value is the y-coordinate of the vertex, which is -1. So, we already know a crucial piece of information about the range of the function!
To really nail down our understanding, let's think about what happens as x moves away from 3. If x is greater than 3, then (x - 3)² will be a positive number that gets larger as x increases. Subtracting 1 from a larger positive number still results in a larger number. Similarly, if x is less than 3, (x - 3)² will also be positive (because squaring a negative number gives you a positive), and again, the function value increases as x moves further away from 3. This confirms that the vertex is indeed the minimum point of the graph, and the parabola opens upwards.
This little analysis gives us a solid foundation for tackling the statements about the graph’s domain and range. Understanding the vertex form and how the parabola behaves around the vertex is crucial for answering questions about quadratic functions. Keep this in mind, and you'll be graphing like a pro in no time!
Analyzing the Domain of the Function
Let's talk domain, guys! The domain of a function is basically all the possible x-values that you can plug into the function and get a real y-value out. In simpler terms, it's what x-values are “allowed” in the function. Now, when we look at Gerald's function, f(x) = (x - 3)² - 1, we need to ask ourselves: are there any restrictions on what x can be?
Think about it this way: Can we plug in any number for x and perform the operations? We're squaring something (which we can always do) and then subtracting 1 (which we can also always do). There’s no division by x, no square roots of potentially negative numbers, and no other funky operations that might limit our x-values. This is a huge clue!
For polynomial functions, like our quadratic here, there are generally no restrictions on the domain. You can plug in any real number, and you'll get a real number out. So, for Gerald’s function, the domain is all real numbers. We can represent this in a few ways:
- Interval Notation: (-∞, ∞) This means all numbers from negative infinity to positive infinity.
- Set-builder Notation: {x | x ∈ ℝ} This reads as “the set of all x such that x is an element of the set of real numbers.”
Now, one of the options states that the domain is {x | x ≥ 3}. This is saying that the domain is only x-values that are greater than or equal to 3. But we just figured out that x can be any real number! So, this statement is definitely false. It's super important to carefully consider what restrictions a function might have and to test specific values if you're unsure. In this case, since there are no restrictions, the domain includes all real numbers, making the statement {x | x ≥ 3} incorrect.
Understanding the domain is a foundational concept in function analysis. It sets the stage for understanding other properties, like the range, intercepts, and overall behavior of the graph. So, nailing down the domain is a crucial first step in understanding any function.
Decoding the Range of the Function
Alright, let's shift our focus to the range! The range of a function is the set of all possible output values, or y-values, that the function can produce. In other words, it’s the “height” of the graph. Remember how we talked about the vertex being a super important point? Well, it’s going to be key for figuring out the range too.
We know Gerald's function is f(x) = (x - 3)² - 1, and we found that the vertex is at (3, -1). Since the coefficient of the (x - 3)² term is positive (it's 1), the parabola opens upwards. This means the vertex is the lowest point on the graph, and -1 is the minimum value of the function. The function can't go any lower than -1.
As x moves away from 3 in either direction (to the left or to the right), the (x - 3)² term becomes larger and larger. This means that the y-values of the function also increase. There's no upper limit to how high the graph can go – it extends upwards infinitely. So, the range includes -1 and all numbers greater than -1.
We can express the range in a couple of ways:
- Interval Notation: [-1, ∞) The square bracket on the -1 indicates that -1 is included in the range, and the parenthesis on infinity indicates that it goes on forever.
- Set-builder Notation: {y | y ≥ -1} This reads as “the set of all y such that y is greater than or equal to -1.”
One of the options given is that the range is {y | y ≥ -1}. This perfectly matches what we just figured out! So, this statement is definitely true. The graph does indeed have a minimum y-value of -1, and it extends upwards from there.
Understanding the range is crucial for fully grasping how a function behaves. It tells us what output values are possible and gives us a sense of the function's limitations. By identifying the vertex and the direction the parabola opens, we can easily determine the range of any quadratic function. Great job, guys!
True Statements About Gerald's Graph
Okay, let's wrap things up and pinpoint the true statements about Gerald's graph of f(x) = (x - 3)² - 1. We've done a thorough analysis, and now it's time to put our knowledge to the test.
We've already established a few key things:
- The vertex is at (3, -1).
- The parabola opens upwards.
- The domain is all real numbers.
- The range is {y | y ≥ -1}.
Based on our analysis, let's revisit the options:
- The domain is {x | x ≥ 3}. False. We know the domain is all real numbers, not just x-values greater than or equal to 3.
- The range is {y | y ≥ -1}. True. This perfectly matches our determination of the range.
So, one of the correct statements is that the range is {y | y ≥ -1}. This makes total sense, given that the parabola opens upwards and the vertex is at (3, -1), making -1 the minimum y-value.
By breaking down the function, identifying its key features, and carefully considering the definitions of domain and range, we've successfully navigated this math problem. Remember, guys, practice makes perfect, and the more you analyze functions like this, the easier it will become! Keep up the awesome work!
Understanding Key Properties of Quadratic Functions and Their Graphs
Domain and Range Explained
Identifying True Statements for Accurate Graph Interpretation
Repair-input-keyword: Which statements are true about the graph of the function f(x)=(x-3)^2-1? Options include domain and range.
Title: Analyzing f(x)=(x-3)^2-1 Identifying Correct Graph Statements