Functions With Maximum, Left, And Down Transformation

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Functions with Maximum, Left, and Down Transformation

Hey guys! Let's dive into this math problem where we're figuring out which functions have a maximum value and are shifted left and down compared to the parent function f(x) = x². It's like we're detectives, but instead of solving crimes, we're solving equations! So, grab your thinking caps, and let's get started.

Understanding the Parent Function: f(x) = x²

First things first, let's make sure we're all on the same page about our starting point: the parent function, f(x) = x². This is a classic parabola, a U-shaped curve that opens upwards. The vertex, or the turning point, is right at the origin (0, 0). This is the lowest point on the graph, meaning the function has a minimum value. Remember this shape, guys, because it's our reference for all the transformations we're about to explore. Understanding this baseline is essential for spotting the shifts and flips in the other functions.

Now, key characteristics of f(x) = x² include:

  • Shape: A parabola opening upwards.
  • Vertex: (0, 0), which is a minimum point.
  • Symmetry: Symmetrical about the y-axis.
  • No Maximum Value: It extends upwards indefinitely.

When we talk about transformations, we're essentially tweaking this basic shape. We can stretch it, compress it, flip it, and slide it around the coordinate plane. Recognizing how each part of a function's equation affects the graph is crucial. For instance, a negative sign in front of the x² term flips the parabola upside down, giving it a maximum value instead of a minimum. Shifts inside the parentheses affect horizontal movement (left or right), while constants added or subtracted outside the parentheses shift the graph vertically (up or down). It's like giving our parabola a makeover and a new home!

Key Transformations to Look For

Before we jump into the answer choices, let's break down what we need to identify. We're on the hunt for functions that meet three specific criteria. These criteria act like clues, helping us narrow down the suspects in our mathematical mystery. Understanding these transformations individually will make it much easier to analyze the given functions.

1. Maximum Value

A function has a maximum value when its graph opens downwards. Think of it as an upside-down U. This happens when the coefficient of the x² term is negative. So, the first thing we need to look for is that negative sign. If there's no negative sign, we can immediately rule out that function. A negative coefficient is like a red flag, telling us this parabola has been flipped.

2. Shifted to the Left

A horizontal shift to the left occurs when we add a constant inside the parentheses with the x-term, like (x + c)², where c is a positive number. Remember, it's the opposite of what you might expect – adding moves it left! This can be a bit tricky, so always pay close attention to the sign inside the parentheses. It's like a secret code, telling us which way the graph has moved horizontally. The larger the value of c, the further left the graph shifts.

3. Shifted Down

A vertical shift downwards happens when we subtract a constant outside the parentheses, like ... - d, where d is a positive number. This one's a bit more intuitive – subtracting pulls the graph down. The bigger the d, the further down the graph goes. So, look for that subtraction sign hanging out at the end of the function. It's like an anchor, pulling the parabola downwards.

Analyzing the Answer Choices

Okay, guys, time to put on our detective hats and examine the suspects! We've got five functions to investigate, and we need to see which ones fit our criteria: maximum value, shifted left, and shifted down. We'll go through each one, one by one, and check it against our checklist of transformations.

A. p(x) = 14(x + 7)² + 1

  • Maximum Value? Nope! The coefficient of (x + 7)² is 14, which is positive. This parabola opens upwards, so it has a minimum value, not a maximum.
  • Shifted Left? Yes! We see (x + 7)², which means it's shifted 7 units to the left.
  • Shifted Down? Nope! We have + 1 at the end, which means it's shifted 1 unit up.

This function is out. It only matches one of our criteria (shifted left), but we need all three. It's like a suspect with a partial alibi – not good enough!

B. q(x) = -5(x + 10)² - 1

  • Maximum Value? Yes! The coefficient of (x + 10)² is -5, which is negative. This parabola opens downwards, so it has a maximum value.
  • Shifted Left? Yes! We see (x + 10)², meaning it's shifted 10 units to the left.
  • Shifted Down? Yes! We have - 1 at the end, so it's shifted 1 unit down.

Bingo! This function hits all three marks. It's got a maximum, it's shifted left, and it's shifted down. This is definitely a function we'll keep in our lineup.

C. s(x) = -(x - 1)² + 0.5

  • Maximum Value? Yes! The coefficient of (x - 1)² is -1 (there's a negative sign), so it has a maximum value.
  • Shifted Left? Nope! We see (x - 1)², which means it's shifted 1 unit to the right.
  • Shifted Down? Nope! We have + 0.5 at the end, meaning it's shifted 0.5 units up.

This function has a maximum value, but it's shifted to the right and up, not left and down. It's close, but no cigar.

D. g(x) = 2x² + 10x - 35

Okay, this one looks a bit different. It's not in the vertex form we've been working with. To figure out the transformations, we need to rewrite it in vertex form, which is a(x - h)² + k, where (h, k) is the vertex of the parabola. This might sound intimidating, but we can do it by completing the square. Think of it as a mathematical makeover, transforming this equation into a form we recognize.

Here's how we complete the square:

  1. Factor out the coefficient of the x² term (which is 2) from the first two terms: 2(x² + 5x) - 35
  2. Take half of the coefficient of the x term (which is 5), square it ((5/2)² = 2.5² = 6.25), and add and subtract it inside the parentheses: 2(x² + 5x + 6.25 - 6.25) - 35
  3. Rewrite the perfect square trinomial: 2((x + 2.5)² - 6.25) - 35
  4. Distribute the 2 and simplify: 2(x + 2.5)² - 12.5 - 35
  5. Combine the constants: 2(x + 2.5)² - 47.5

Now we have the function in vertex form: g(x) = 2(x + 2.5)² - 47.5

  • Maximum Value? Nope! The coefficient of (x + 2.5)² is 2, which is positive. This parabola opens upwards.
  • Shifted Left? Yes! We see (x + 2.5)², meaning it's shifted 2.5 units to the left.
  • Shifted Down? Yes! We have - 47.5 at the end, so it's shifted 47.5 units down.

Even though it's shifted left and down, it doesn't have a maximum value. So, it doesn't meet all our criteria.

E. t(x) = -2x² - 4x - 3

Let's do the same thing for this one – complete the square to get it into vertex form:

  1. Factor out the -2: -2(x² + 2x) - 3
  2. Take half of the coefficient of the x term (which is 2), square it (1² = 1), and add and subtract it inside the parentheses: -2(x² + 2x + 1 - 1) - 3
  3. Rewrite the perfect square trinomial: -2((x + 1)² - 1) - 3
  4. Distribute the -2 and simplify: -2(x + 1)² + 2 - 3
  5. Combine the constants: -2(x + 1)² - 1

So, t(x) = -2(x + 1)² - 1

  • Maximum Value? Yes! The coefficient of (x + 1)² is -2, which is negative. It has a maximum.
  • Shifted Left? Yes! We see (x + 1)², so it's shifted 1 unit to the left.
  • Shifted Down? Yes! We have - 1 at the end, so it's shifted 1 unit down.

This one's a winner! It's got a maximum value, and it's shifted left and down.

Conclusion: The Functions That Fit the Bill

Alright, guys, after our thorough investigation, we've found the functions that have a maximum value and are transformed to the left and down of the parent function, f(x) = x². The functions that fit the criteria are:

  • B. q(x) = -5(x + 10)² - 1
  • E. t(x) = -2(x + 1)² - 1

We did it! By breaking down the transformations and carefully analyzing each function, we were able to solve the problem. Remember, math isn't about memorizing formulas; it's about understanding the concepts and applying them step by step. Keep practicing, and you'll become transformation detectives in no time!