Functions With Range Y ≤ 5: Solving Quadratic Equations

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Hey guys! Today, we're diving into a fun little math problem that involves figuring out which quadratic function has a range where y is always less than or equal to 5. This might sound a bit technical, but don't worry, we'll break it down step by step. We'll explore what the range of a function means, how it relates to the graph of a quadratic equation, and then nail down the correct answer. So, grab your thinking caps, and let's get started!

Understanding the Range of a Function

First off, let's talk about what the range of a function actually means. Think of a function like a machine: you put something in (the input, usually x), and it spits something out (the output, usually y or f(x)). The range is basically the set of all possible y-values that the function can produce. In simpler terms, it's all the possible heights the graph of the function can reach on the vertical axis.

Now, when we're dealing with quadratic functions – those with an x² term – their graphs are parabolas, which look like U-shaped curves. These parabolas can open upwards or downwards. If the parabola opens upwards, it has a minimum point, and the range includes all y-values greater than or equal to that minimum. If the parabola opens downwards, it has a maximum point, and the range includes all y-values less than or equal to that maximum. This is key to solving our problem.

For a parabola defined by the equation f(x) = a(x - h)² + k, the vertex (the minimum or maximum point) is at the point (h, k). The value of a tells us whether the parabola opens upwards (a > 0) or downwards (a < 0). If a is positive, k is the minimum y-value, and if a is negative, k is the maximum y-value. Got it? Great! Let's move on to our specific problem.

Analyzing the Given Functions

Okay, so we need to find the function with a range of y ≤ 5. This tells us that the parabola must open downwards (because the y-values are less than or equal to a certain number), and the maximum y-value (the k value of the vertex) must be 5. Let's look at the options we have:

A. f(x) = (x - 4)² + 5 B. f(x) = -(x - 4)² + 5 C. f(x) = (x - 5)² + 4 D. f(x) = -(x - 5)² + 4

Let's break down each option and see if it fits the bill:

  • Option A: f(x) = (x - 4)² + 5
    • Here, a = 1 (which is positive), so the parabola opens upwards. This means it has a minimum value, not a maximum. So, this isn't our answer.
  • Option B: f(x) = -(x - 4)² + 5
    • Here, a = -1 (which is negative), so the parabola opens downwards. The vertex is at (4, 5), meaning the maximum y-value is 5. This looks promising!
  • Option C: f(x) = (x - 5)² + 4
    • Here, a = 1 (positive), so the parabola opens upwards. Again, this isn't what we're looking for.
  • Option D: f(x) = -(x - 5)² + 4
    • Here, a = -1 (negative), so the parabola opens downwards. However, the vertex is at (5, 4), meaning the maximum y-value is 4, not 5. So, this isn't our answer either.

Identifying the Correct Answer

Alright, after carefully analyzing each option, it's pretty clear that Option B is the winner. The function f(x) = -(x - 4)² + 5 has a negative coefficient for the x² term, which means the parabola opens downwards. The vertex is at the point (4, 5), indicating that the maximum y-value is 5. Therefore, the range of this function is indeed y ≤ 5, exactly what we were looking for!

Graphing for Visual Confirmation

Sometimes, a visual aid can really solidify our understanding. If we were to graph these functions, we'd see that:

  • Option A would be a parabola opening upwards with its lowest point at y = 5.
  • Option B would be a parabola opening downwards with its highest point at y = 5. Bingo!
  • Option C would be a parabola opening upwards with its lowest point at y = 4.
  • Option D would be a parabola opening downwards with its highest point at y = 4.

Seeing the graphs can make it super clear why Option B is the only one that fits our criteria.

Key Concepts Recap

Before we wrap up, let's quickly recap the main concepts we used to solve this problem. This will help you tackle similar questions in the future:

  • Range of a function: The set of all possible output values (y-values).
  • Quadratic functions and parabolas: Functions with an x² term create U-shaped graphs called parabolas.
  • Vertex form of a quadratic: f(x) = a(x - h)² + k, where (h, k) is the vertex.
  • Direction of opening: If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
  • Maximum and minimum values: For a parabola opening downwards, k is the maximum y-value; for a parabola opening upwards, k is the minimum y-value.

Practice Makes Perfect

Math is like learning a new language – the more you practice, the better you get! Try working through similar problems, maybe changing the range or the vertex of the parabola. The key is to understand how the different parts of the quadratic equation affect the graph and the range of the function.

Conclusion: Mastering Quadratic Ranges

So, there you have it! We've successfully identified the function with a range of y ≤ 5. By understanding the range of a function, how it relates to the graph of a quadratic equation, and the significance of the vertex, we were able to confidently choose the correct answer. Remember, guys, the key to mastering these concepts is practice and a solid understanding of the fundamentals. Keep up the great work, and you'll be solving these problems like a pro in no time! Now you know how to confidently tackle quadratic function range questions. Keep practicing and you'll be a parabola pro in no time!