Equation Above X-axis? Find The Graph!

Hey math enthusiasts! Ever wondered which equations create graphs that just hang out above the x-axis, never dipping below? It's a cool concept, and today, we're diving deep into how to spot these equations. We'll break down the options, making sure you not only get the right answer but also understand the why behind it. So, let's get started and unravel this mathematical mystery together!

Understanding the Question

Okay, before we jump into the options, let's make sure we're all on the same page. The question asks: Which equation has a graph that lies entirely above the x-axis? What does this really mean? Essentially, we're looking for a graph where the y-values are always positive. Think of it like this: the graph is like a bird flying, and the x-axis is the ground. We want to find the bird that never touches the ground.

To visualize this, imagine a coordinate plane. The x-axis is that horizontal line running across. A graph that lies entirely above it means every single point on that graph has a y-value greater than zero. No dips into the negative zone! This is super important because it helps us narrow down our choices. We need an equation that, no matter what x-value we plug in, will always spit out a positive y-value. This usually involves understanding how different parts of an equation affect its graph, particularly when we're dealing with quadratic equations.

Now, let's consider what types of equations might do this. Linear equations (straight lines) can cross the x-axis unless they're perfectly horizontal and above the axis. But quadratic equations, with their U-shaped parabolas, offer more interesting possibilities. A parabola that opens upwards and has its vertex (the lowest point) above the x-axis will fit our criteria perfectly. So, we're on the hunt for an upward-opening parabola that never touches or crosses that x-axis. With this understanding, we can start analyzing the given options and see which one matches this description.

Analyzing Option A: y = -(x + 7)² + 7

Let's kick things off with Option A: y = -(x + 7)² + 7. At first glance, this equation might seem a bit intimidating, but let’s break it down step by step. This is a quadratic equation, which means its graph will be a parabola. The general form of a quadratic equation in vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola, and 'a' determines whether the parabola opens upwards or downwards.

In our case, we have a negative sign in front of the squared term: -(x + 7)². This negative sign is crucial because it tells us that 'a' is negative (a = -1). And what does a negative 'a' mean? It means the parabola opens downwards. Imagine a frown instead of a smile. This is a big clue because if a parabola opens downwards, it has a maximum point (the vertex) rather than a minimum. If this maximum point is above the x-axis, the parabola will eventually cross the x-axis as it extends downwards on both sides.

Now, let’s find the vertex of this parabola. Looking at our equation y = -(x + 7)² + 7, we can identify h and k. Remember, the vertex form is y = a(x - h)² + k, so we have h = -7 and k = 7. This means the vertex of our parabola is at the point (-7, 7). Okay, the y-coordinate of the vertex is 7, which is positive, so the vertex itself is above the x-axis. But, and this is a big but, because the parabola opens downwards, it will definitely cross the x-axis. Think of it like a hill – it goes up to the peak (the vertex) and then slopes down on both sides.

To really drive this point home, we can think about what happens as x gets very large or very small (approaching positive or negative infinity). The term -(x + 7)² will become a very large negative number, and adding 7 won't be enough to keep y positive. So, the graph will indeed go below the x-axis. Therefore, Option A is not the equation we’re looking for. It's a downward-opening parabola with a vertex above the x-axis, but it doesn't stay entirely above the x-axis. We need a parabola that opens upwards and has its vertex above the x-axis. Let's move on to the next option!

Analyzing Option B: y = (x - 7)² - 7

Alright, let’s tackle Option B: y = (x - 7)² - 7. Just like Option A, this is another quadratic equation, meaning we're dealing with a parabola. But there's a key difference here that we need to spot right away. Notice that the term in front of the squared part, (x - 7)², is positive (there's an implied +1). This is super important because it tells us the parabola opens upwards. Think of it as a smile – a U-shape that extends upwards to infinity.

So, what does an upward-opening parabola mean for our quest to find a graph that stays entirely above the x-axis? It means we're on the right track! An upward-opening parabola has a minimum point, called the vertex. If this vertex is above the x-axis, the entire parabola will be above the x-axis. But if the vertex is below the x-axis, the parabola will cross the x-axis at some point. Our mission now is to find the vertex of this parabola and see where it sits.

Using the vertex form of a quadratic equation, y = a(x - h)² + k, we can identify the vertex. In our equation, y = (x - 7)² - 7, we can see that h = 7 and k = -7. This means the vertex is at the point (7, -7). Uh oh! Notice that the y-coordinate of the vertex is -7, which is negative. This is a red flag.

What does a vertex at (7, -7) tell us? It tells us that the lowest point of the parabola is below the x-axis. Since the parabola opens upwards, it will indeed cross the x-axis at two points. Imagine drawing this parabola – it starts below the x-axis, curves upwards, crosses the x-axis, reaches its minimum point at (7, -7), and then curves back up, crossing the x-axis again on the other side. This mental picture helps solidify why Option B isn't our answer.

To summarize, Option B is an upward-opening parabola, which is good, but its vertex is below the x-axis, which means it will cross the x-axis. We need a parabola that opens upwards and has its vertex above the x-axis. So, Option B is out. Let's keep searching for the equation that keeps its graph entirely above the x-axis!

Analyzing Option C: y = (x - 7)² + 7

Let's dive into Option C: y = (x - 7)² + 7. This equation looks quite similar to Option B, but there's a crucial difference that will make all the difference in the world. Just like Option B, this is a quadratic equation, and the term in front of the squared part, (x - 7)², is positive (again, an implied +1). This means we're dealing with an upward-opening parabola – a happy, smiling U-shape.

As we discussed earlier, an upward-opening parabola has a minimum point, the vertex. And for the graph to stay entirely above the x-axis, this vertex needs to be above the x-axis as well. So, let's pinpoint the vertex of this parabola and see where it lies. Using the vertex form y = a(x - h)² + k, we can identify h and k in our equation y = (x - 7)² + 7. We see that h = 7 and k = 7. This means the vertex is at the point (7, 7).

Now, this is exciting! The vertex is at (7, 7), which means its y-coordinate is 7, a positive number. This tells us that the lowest point of the parabola is above the x-axis. And since the parabola opens upwards, it will never cross or touch the x-axis. Picture this: a U-shaped curve sitting entirely above the horizontal line of the x-axis. It's like a bridge that never touches the water below.

To really solidify this, let's think about what happens to the y-value as x gets very large or very small. The term (x - 7)² will always be positive or zero (since anything squared is non-negative). Adding 7 to it ensures that y will always be greater than or equal to 7. There's no way for y to become negative. This is exactly what we're looking for – an equation where the graph stays entirely above the x-axis.

So, Option C is looking like a very strong contender! It's an upward-opening parabola with a vertex above the x-axis. But, just to be absolutely sure, let's quickly analyze Option D to make sure it doesn't fit the bill even better.

Analyzing Option D: y = (x - 7)²

Finally, let's examine Option D: y = (x - 7)². This equation is another quadratic, and like Options B and C, the term in front of the squared part, (x - 7)², is positive (an implied +1). This means we're dealing with an upward-opening parabola – a U-shaped graph that smiles at us.

We know that for a parabola to lie entirely above the x-axis, it needs to open upwards and have its vertex above the x-axis. So, let's find the vertex of this parabola. Using the vertex form y = a(x - h)² + k, we can identify h and k in our equation y = (x - 7)². Here, h = 7, and since there's no constant term added or subtracted, k = 0. This means the vertex is at the point (7, 0).

Okay, the vertex is at (7, 0). Notice anything special about this point? It lies on the x-axis! The y-coordinate is zero. This is a crucial observation. It means the parabola touches the x-axis at its lowest point. While the rest of the parabola will be above the x-axis, it doesn't stay entirely above it. It has that one point of contact, that single moment where it dips down and kisses the x-axis before curving back up.

Imagine drawing this parabola – it’s a U-shape that just barely touches the x-axis at the point (7, 0). It's close to being entirely above, but that one touch disqualifies it from our search. We need an equation where the graph never touches the x-axis.

To further understand this, let's think about the y-values. The term (x - 7)² will always be positive or zero. It can be zero when x = 7, which is why the parabola touches the x-axis at (7, 0). But we need y to be strictly greater than zero for the graph to stay entirely above the x-axis. So, Option D, while close, doesn't quite make the cut. It touches the x-axis, and we need a graph that avoids it completely.

Conclusion: The Equation That Stays Above

Alright, guys, we've journeyed through all the options, dissected parabolas, and hunted for that elusive equation whose graph lives entirely above the x-axis. Let's recap our findings:

• Option A: y = -(x + 7)² + 7 – This was a downward-opening parabola, so it definitely crosses the x-axis.
• Option B: y = (x - 7)² - 7 – This was an upward-opening parabola, but its vertex was below the x-axis, so it also crosses the x-axis.
• Option C: y = (x - 7)² + 7 – This was an upward-opening parabola, and its vertex was above the x-axis, ensuring the entire graph stays above the x-axis. This is our winner!
• Option D: y = (x - 7)² – This was an upward-opening parabola, but its vertex was on the x-axis, meaning it touches the x-axis, which we don't want.

So, the answer we've been searching for is Option C. The equation y = (x - 7)² + 7 has a graph that lies entirely above the x-axis. It's an upward-opening parabola with its vertex comfortably perched above the x-axis, never daring to dip below.

We've not only found the correct answer, but we've also explored the reasoning behind it. We've seen how the sign in front of the squared term determines whether a parabola opens upwards or downwards, and how the vertex plays a crucial role in determining whether a graph stays above or crosses the x-axis. This understanding is key to tackling similar problems in the future. Keep practicing, keep exploring, and you'll become a math whiz in no time! Great job, everyone!