Gráfica De Parábola: H(x) = -x² + 4x + 5

Hey guys, let's dive into the awesome world of parabolas and graph the function $h(x) = -x^2 + 4x + 5$. Understanding how to graph these mathematical curves is super useful, especially when you're dealing with real-world problems that can be modeled by quadratic equations. Think projectile motion, optimizing areas, or even understanding the trajectory of a ball. So, grab your notebooks, pens, and let's get this done!

Understanding the Equation: What's a Parabola Anyway?

Alright, first things first, what exactly is a parabola? In the realm of mathematics, a parabola is a symmetrical U-shaped curve. It's defined as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed straight line (the directrix). When we talk about graphing a parabola in this context, we're usually dealing with a quadratic function, which has the general form $f(x) = ax^2 + bx + c$. The key feature of our equation, $h(x) = -x^2 + 4x + 5$, is the $x^2$ term. The coefficient of this term, in our case $-1$, tells us a lot about the shape and direction of our parabola. Since the coefficient 'a' is negative ($a = -1$), our parabola will open downwards. Imagine a frown! This means it will have a maximum point, which we call the vertex. If 'a' were positive, it would open upwards, like a smile, and have a minimum point at its vertex. So, for $h(x) = -x^2 + 4x + 5$, we already know it's going to be an upside-down U-shape. This is crucial information as we start plotting points and sketching our graph.

Finding the Vertex: The Peak of Our Parabola

Now, let's talk about the vertex. This is arguably the most important point on our parabola because it represents the maximum (or minimum, if it opened upwards) value of the function. For a quadratic equation in the form $h(x) = ax^2 + bx + c$, the x-coordinate of the vertex can be found using the formula $x = -b / (2a)$. Let's plug in our values from $h(x) = -x^2 + 4x + 5$. Here, $a = -1$, $b = 4$, and $c = 5$. So, the x-coordinate of the vertex is $x = -(4) / (2 * -1) = -4 / -2 = 2$. Boom! We've found the x-value where our parabola reaches its peak. To find the corresponding y-coordinate (which is the actual maximum height in this case), we just substitute this x-value back into our original function: $h(2) = -(2)^2 + 4(2) + 5 = -4 + 8 + 5 = 9$. So, the vertex of our parabola is at the point (2, 9). This point is super important – it's the highest point on our graph and gives us a anchor for our sketch. Remember, since 'a' is negative, this is indeed a maximum point.

Determining the Axis of Symmetry: The Mirror Line

Another super helpful feature of a parabola is its axis of symmetry. This is a vertical line that passes through the vertex and divides the parabola into two perfectly symmetrical halves. Whatever happens on one side of the axis of symmetry is mirrored exactly on the other side. For any quadratic function, the axis of symmetry is simply the vertical line $x = -b / (2a)$, which, conveniently, is the same x-coordinate we found for the vertex! In our case, with the vertex at $x=2$, the axis of symmetry is the line x = 2. This line is like a mirror for our parabola. If you were to fold the graph along this line, the two sides would perfectly overlap. Knowing the axis of symmetry helps us plot points accurately. Once we find a point on one side of the axis, we automatically know its corresponding symmetrical point on the other side. For instance, if we find a point 1 unit to the right of the axis of symmetry, there's an identical point 1 unit to the left at the same height. This makes graphing so much easier and ensures our parabola is perfectly balanced and symmetrical as it should be.

Finding the y-intercept: Where the Parabola Crosses the y-axis

The y-intercept is another critical point to identify when graphing a parabola. This is the point where the graph crosses the y-axis. Mathematically, this occurs when $x = 0$. So, to find the y-intercept for our function $h(x) = -x^2 + 4x + 5$, we simply substitute $x = 0$ into the equation: $h(0) = -(0)^2 + 4(0) + 5 = 0 + 0 + 5 = 5$. Therefore, the y-intercept is at the point (0, 5). This is the point where our downward-opening parabola hits the vertical axis. It's a straightforward calculation, but it gives us another concrete point to plot. Seeing where the graph intersects the y-axis is important for understanding the overall position and behavior of the parabola on the coordinate plane. It's another piece of the puzzle that helps us draw an accurate representation of our function.

Finding the x-intercepts (Roots): Where the Parabola Hits the x-axis

The x-intercepts, also known as the roots or zeros of the quadratic equation, are the points where the parabola crosses the x-axis. At these points, the value of $h(x)$ is equal to 0. So, to find the x-intercepts for $h(x) = -x^2 + 4x + 5$, we need to solve the equation $-x^2 + 4x + 5 = 0$. There are a few ways to do this: factoring, completing the square, or using the quadratic formula. Let's try factoring first, as it's often the quickest if it works. We are looking for two numbers that multiply to $-5$ (the constant term) and add up to $4$ (the coefficient of the x term). These numbers are $5$ and $-1$. So, we can rewrite the equation as $(x + 5)( -x + 1) = 0$. Wait, that's not quite right. Let's factor out a $-1$ first to make the leading coefficient positive: $-1(x^2 - 4x - 5) = 0$. Now, we need two numbers that multiply to $-5$ and add to $-4$. Those numbers are $-5$ and $1$. So, we get $-1(x - 5)(x + 1) = 0$. For this equation to be true, either $(x - 5) = 0$ or $(x + 1) = 0$. Solving these, we get $x = 5$ and $x = -1$. So, the x-intercepts are at the points (5, 0) and (-1, 0). These are the points where our parabola touches the horizontal axis. If factoring seems tricky, you can always use the quadratic formula: $x = [-b ± \sqrt{b^2 - 4ac}] / (2a)$. Plugging in our values $a=-1, b=4, c=5$: $x = [-4 ± \sqrt{4^2 - 4(-1)(5)}] / (2(-1)) = [-4 ± \sqrt{16 + 20}] / (-2) = [-4 ± \sqrt{36}] / (-2) = [-4 ± 6] / (-2)$. This gives us two solutions: $x1 = (-4 + 6) / (-2) = 2 / -2 = -1$ and $x2 = (-4 - 6) / (-2) = -10 / -2 = 5$. Both methods confirm our x-intercepts!

Plotting the Points and Sketching the Graph

Okay, guys, we've gathered all the crucial information:

• Vertex: (2, 9)
• Axis of Symmetry: x = 2
• y-intercept: (0, 5)
• x-intercepts: (-1, 0) and (5, 0)

Now comes the fun part – plotting these points on a graph and sketching the parabola! Grab your graph paper or open up your favorite graphing tool. First, draw your x and y axes. Label them appropriately. Now, plot the vertex (2, 9). This is the highest point. Next, plot the y-intercept (0, 5). Remember the axis of symmetry (x = 2)? Since the y-intercept is 2 units to the left of the axis of symmetry (0 is 2 less than 2), its symmetrical point will be 2 units to the right of the axis of symmetry, at the same height. So, plot another point at (4, 5). Now, plot the x-intercepts (-1, 0) and (5, 0). These are where the parabola crosses the x-axis. We have five key points: (-1, 0), (0, 5), (2, 9), (4, 5), and (5, 0).

With these points plotted, you can start sketching the curve. Remember that parabolas are smooth, continuous curves. Start from one of the x-intercepts, curve upwards towards the y-intercept, continue curving up to the vertex, and then curve downwards symmetrically through the other y-intercept and down towards the other x-intercept (or beyond, if you were to extend the graph). Since our parabola opens downwards ($a = -1$), the curve should be U-shaped, opening downwards, with its peak at the vertex. Make sure the curve is symmetrical with respect to the line $x = 2$. Don't draw sharp corners; keep it smooth and flowing. The shape should resemble a frown. The points you've plotted should lie perfectly on this smooth curve. You can always find more points by picking an x-value, calculating $h(x)$, and plotting it, but with the vertex, intercepts, and symmetry, you've got more than enough to draw a very accurate sketch.

Conclusion: Mastering Parabola Graphs

And there you have it, guys! We've successfully graphed the parabola represented by the function $h(x) = -x^2 + 4x + 5$. By breaking it down into finding the vertex, axis of symmetry, y-intercept, and x-intercepts, we were able to accurately plot and sketch this quadratic curve. Remember, the coefficient 'a' dictates whether the parabola opens up or down, the vertex is your highest or lowest point, the axis of symmetry is your mirror line, and the intercepts show where it crosses the axes. Practicing with different quadratic equations will make you a pro at graphing parabolas. These skills are not just for math class; they're foundational for understanding many real-world phenomena. Keep practicing, and you'll be graphing parabolas like a champ in no time! Happy graphing!