Finding The Minimum Value Of G(x) Step By Step Solution

Hey guys! Let's dive into a fun math problem today. We're going to explore how to find the minimum value of a function, which might sound intimidating, but trust me, we'll break it down into easy steps. Our mission is to figure out at what value of x the function g(x) hits its lowest point. So, grab your thinking caps, and let's get started!

Understanding the Functions: f(x) and g(x)

Before we can even think about g(x), we need to understand what f(x) is all about. We're given that f(x) = 4x² + 64x + 262. This is a quadratic function, and quadratic functions have a beautiful U-shape (or an upside-down U if the coefficient of the x² term is negative). Because the coefficient of our x² term (which is 4) is positive, we know our parabola opens upwards, meaning it has a minimum point – exactly what we're looking for!

Now, let's talk about g(x). We're told that g(x) = f(x + 5). What does this mean? It means we're taking our original function f(x) and plugging in (x + 5) wherever we see x. This is called a horizontal translation. Imagine taking the graph of f(x) and sliding it left or right. That's exactly what's happening here. Specifically, adding 5 inside the function shifts the graph 5 units to the left. Understanding this shift is key to solving the problem without doing a ton of calculations. So, we have a quadratic function f(x), and g(x) is just a shifted version of it. Both will have a minimum point, and figuring out how the shift affects that minimum is our goal.

Finding the Minimum of f(x)

Let's first focus on f(x). There are a couple of ways we can find the minimum point of a quadratic function. One way is to complete the square. This involves rewriting the quadratic in a form that makes the vertex (the minimum or maximum point) easily visible. The other way, which is often faster, is to use the formula for the x-coordinate of the vertex of a parabola. Remember that the vertex form of a quadratic equation is f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The x-coordinate of the vertex, h, can be found using the formula h = -b / 2a, where a and b are the coefficients from the standard form of the quadratic equation, f(x) = ax² + bx + c. This is a super useful formula to remember!

In our case, f(x) = 4x² + 64x + 262, so a = 4 and b = 64. Let's plug these values into our formula: h = -64 / (2 * 4) = -64 / 8 = -8. So, the x-coordinate of the minimum point of f(x) is -8. This means that f(x) reaches its minimum value when x = -8. We're halfway there! We know where f(x) hits its low point, now we need to relate this to g(x).

Connecting f(x) and g(x): The Horizontal Shift

Remember that g(x) = f(x + 5)? This means that the graph of g(x) is the graph of f(x) shifted 5 units to the left. Think about it this way: if f(x) reaches its minimum at x = -8, then g(x) will reach its minimum at a value 5 less than -8. This is because whatever input minimized f(x), we now need an input for g(x) that, after adding 5, will give us that same input. To put it simply, if f(-8) is the minimum value, then we want to find x such that x + 5 = -8. This is the heart of the problem – understanding how the horizontal shift affects the minimum point.

To find where g(x) reaches its minimum, we simply subtract 5 from the x-coordinate of the minimum of f(x). So, if f(x) is minimized at x = -8, then g(x) is minimized at x = -8 - 5 = -13. That's it! We've found the value of x that minimizes g(x) by understanding the relationship between f(x) and g(x) and how horizontal shifts work. It's like detective work, piecing together the clues to solve the mystery.

Why the Other Options are Incorrect

Let's briefly think about why the other answer choices are incorrect. We found that the minimum of g(x) occurs at x = -13. This was determined by first finding the minimum of f(x) at x = -8 and then accounting for the horizontal shift of 5 units to the left caused by the f(x + 5) transformation. If we didn't account for the shift, we might mistakenly think the minimum occurs at x = -8, which is the minimum of f(x), not g(x). The other options, x = -5 and x = -3, don't logically follow from the shift. x = -5 might seem tempting because of the “+5” in f(x + 5), but remember that adding inside the function shifts the graph left, not right. So, these options are just distractions from the correct application of the shift.

The Answer

So, putting it all together, the function g(x) reaches its minimum when x = -13. The correct answer is A. -13. We navigated through the problem by understanding quadratic functions, horizontal shifts, and how to find the minimum point. Great job, guys! You've conquered a potentially tricky problem by breaking it down into manageable steps. Remember, math problems are just puzzles waiting to be solved!

Understanding Quadratic Functions and Minimum Values

To effectively solve this mathematical challenge, let's begin by clarifying the core concepts surrounding quadratic functions and their minimum values. Quadratic functions, typically represented in the form f(x) = ax² + bx + c, describe a parabola when graphed. These parabolas exhibit a distinct U-shape if the coefficient a is positive, or an inverted U-shape if a is negative. Our function, f(x) = 4x² + 64x + 262, clearly displays a positive leading coefficient (a = 4), indicating that it opens upwards, and therefore, possesses a minimum point. This minimum point is also known as the vertex of the parabola.

The Significance of the Vertex

The vertex plays a crucial role in identifying the minimum value of a quadratic function. Understanding how to locate this vertex is essential for solving a variety of problems related to optimization and function behavior. For a parabola that opens upwards, the vertex represents the lowest point on the graph, thus indicating the minimum value of the function. The x-coordinate of the vertex tells us the value of x at which this minimum occurs, and the y-coordinate (the function's value at that x) gives us the minimum value itself. To find this vertex, mathematicians have developed several methods, including completing the square and using a specific formula derived from calculus principles. The most straightforward approach for our case involves using the formula x = -b / 2a, which directly calculates the x-coordinate of the vertex based on the coefficients of the quadratic equation. This method provides a quick and efficient way to determine where the minimum of our function f(x) occurs.

Horizontal Transformations and Their Impact

Now, let's delve into the concept of horizontal transformations and how they influence the graph of a function. In our problem, we're given that g(x) = f(x + 5). This expression denotes a horizontal transformation applied to the original function f(x). Specifically, replacing x with (x + 5) inside the function shifts the entire graph 5 units to the left along the x-axis. This is a fundamental concept in function transformations, where adding a constant to x inside the function moves the graph horizontally, and the direction is opposite to the sign of the constant. Therefore, a “+5” implies a shift to the left. Understanding this transformation is key to predicting how the minimum point of f(x) will change when we consider g(x). If we know where f(x) reaches its minimum, we can then easily determine where g(x) will be minimized by accounting for this horizontal shift. Essentially, the transformation alters the x-coordinate at which the minimum occurs, while the minimum value itself (the y-coordinate) remains unchanged.

Determining the Minimum Value of f(x) = 4x² + 64x + 262

To proceed, we need to pinpoint the minimum value of our original function, f(x) = 4x² + 64x + 262. As previously discussed, this involves finding the x-coordinate of the vertex of the parabola represented by this function. Utilizing the formula x = -b / 2a, we can easily calculate this value. Here, a = 4 and b = 64, so substituting these values into the formula gives us x = -64 / (2 * 4) = -64 / 8 = -8. This result tells us that f(x) reaches its minimum when x = -8. This is a critical piece of information that we will use to determine the minimum of g(x). The x-coordinate of the vertex for f(x) provides the baseline from which we will adjust based on the horizontal shift introduced in the definition of g(x). It’s important to recognize that this x-value is not the final answer but rather a stepping stone towards understanding the behavior of g(x).

Verifying the Minimum Value

While the formula x = -b / 2a efficiently provides the x-coordinate of the vertex, it's also helpful to conceptually verify why this works. The vertex represents the point where the parabola changes direction—from decreasing to increasing (for a parabola opening upwards) or vice versa. This transition occurs at the axis of symmetry of the parabola, which is a vertical line passing through the vertex. The formula x = -b / 2a essentially calculates the equation of this line of symmetry. Therefore, by finding x = -8, we've identified the axis of symmetry for f(x), confirming that this is indeed the x-coordinate where the function reaches its minimum. To fully illustrate this, imagine the parabola folding along this line; the two halves would perfectly overlap, highlighting the symmetrical nature around the vertex. This conceptual understanding strengthens our grasp of the mathematics involved and provides a solid foundation for tackling more complex problems.

Calculating the Minimum for g(x) = f(x + 5)

Now comes the crucial step: determining where g(x) = f(x + 5) reaches its minimum. We've already established that g(x) represents a horizontal shift of f(x) by 5 units to the left. This means that if f(x) reaches its minimum at x = -8, then g(x) will reach its minimum at an x-value that is 5 units less than -8. This shift is a direct consequence of replacing x with (x + 5) in the function. To find this new minimum point, we simply subtract 5 from the minimum point of f(x): -8 - 5 = -13. Therefore, g(x) reaches its minimum when x = -13. This is the core of the solution – understanding and applying the horizontal shift transformation to the minimum point of the original function.

Final Verification and Conclusion

To ensure our answer is correct, we can conceptually think through the transformation again. The graph of g(x) is simply f(x) moved 5 units to the left. Thus, the x-coordinate where the minimum occurs will also shift 5 units to the left. Since the minimum of f(x) is at x = -8, the minimum of g(x) must be at x = -8 - 5 = -13. This confirms our calculation. Therefore, the value of x for which g(x) reaches its minimum is x = -13. The correct answer is A. -13. This methodical approach, combining algebraic calculation with conceptual understanding, is vital for confidently tackling similar mathematical challenges. By breaking down the problem into manageable parts—identifying the type of function, finding the minimum of the original function, understanding horizontal transformations, and applying the transformation to the minimum—we've successfully solved the problem. Remember, guys, practice and a clear understanding of underlying concepts are key to success in mathematics.

Why x = -13 is the Definitive Answer

To further solidify our understanding, let's reiterate why x = -13 is the correct answer and why other options are incorrect. The fundamental reason lies in the horizontal shift applied to the function. By transforming f(x) into f(x + 5), we're essentially asking: “What value of x, when added to 5, will give us the same input that minimized f(x)?” We know that f(x) is minimized at x = -8. Therefore, we need to find x such that x + 5 = -8. Solving this equation, we indeed find x = -13. This logical pathway highlights the direct relationship between the minimum points of f(x) and g(x), dictated by the transformation. The options other than -13 do not satisfy this fundamental relationship, hence their incorrectness.

Stepping Away from Guesswork: The Importance of Process

Instead of resorting to guesswork, our approach has been rooted in a clear, step-by-step process. We first established the nature of the quadratic function and the significance of its vertex. Then, we tackled the concept of horizontal transformations, understanding their impact on the function's graph. We calculated the minimum point of f(x) using a reliable formula and applied the horizontal shift to determine the minimum point of g(x). This structured methodology is crucial for problem-solving in mathematics, where understanding the underlying principles is far more valuable than simply memorizing formulas. By following this approach, you empower yourselves to confidently solve a wider array of problems, even those with slight variations or added complexities. Remember, the journey through the problem is just as important as the destination.