Graphing The Function Y = X^2 - 5 A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of functions, specifically the quadratic function y = x^2 - 5. We'll tackle how to graph this function over a defined interval, which in our case is $-3 ext{ less than or equal to } x ext{ less than or equal to } 3$. This means we're only looking at the part of the graph where x falls between -3 and 3, inclusive. So, grab your pencils, and let's get started!

Step 1 Fill in the Table of Values

First things first, we need to figure out what the y-values are for different x-values within our interval. To do this, we'll use a table. Think of it as our function's personal assistant, keeping track of the inputs (x) and their corresponding outputs (y). We’ve got a handy table already set up, and our job is to plug in each x-value into the function y = x^2 - 5 and calculate the y-value. This is where the magic happens, guys! Let’s break it down:

• When x = -3: Substitute -3 into the equation: y = (-3)^2 - 5 = 9 - 5 = 4. So, when x is -3, y is 4. We've got our first pair!
• When x = -2: Substitute -2 into the equation: y = (-2)^2 - 5 = 4 - 5 = -1. Nice! When x is -2, y is -1.
• When x = -1: Substitute -1: y = (-1)^2 - 5 = 1 - 5 = -4. Okay, x = -1 gives us y = -4.
• When x = -0.5: Substitute -0.5: y = (-0.5)^2 - 5 = 0.25 - 5 = -4.75. Getting a bit more precise here!
• When x = 0: Substitute 0: y = (0)^2 - 5 = 0 - 5 = -5. Ah, the simplest calculation! When x is 0, y is -5. This is a crucial point – the y-intercept!
• When x = 0.5: Substitute 0.5: y = (0.5)^2 - 5 = 0.25 - 5 = -4.75. Notice that this is the same y-value as when x was -0.5. Interesting pattern!
• When x = 1: Substitute 1: y = (1)^2 - 5 = 1 - 5 = -4. Another value we've seen before! When x is 1, y is -4.
• When x = 2: Substitute 2: y = (2)^2 - 5 = 4 - 5 = -1. Yep, this matches the y-value when x was -2.
• When x = 3: Substitute 3: y = (3)^2 - 5 = 9 - 5 = 4. And finally, this matches the y-value when x was -3.

Now, let's put all those calculated y-values into our table. Filling this table correctly is essential because these pairs of (x, y) values will be the points we plot on our graph. If we mess up the calculations here, our graph won't represent the function accurately. Double-check your work, guys! It’s always better to be safe than sorry when dealing with math. We want our graph to be the star of the show, a perfect representation of y = x^2 - 5, and that starts with accurate calculations.

Step 2 Plotting the Graph of the Function

Alright, with our table brimming with (x, y) pairs, it's time to transform these numbers into a visual masterpiece: the graph of our function y = x^2 - 5. This is where we'll see the shape of the function come to life. Think of plotting points as connecting the dots, but instead of making a picture of a cat or a house, we're drawing the essence of a mathematical relationship.

First, we need our stage – the coordinate plane. You know, the one with the x-axis (horizontal) and the y-axis (vertical). These axes are our guides, helping us pinpoint the exact location of each point. Remember, each (x, y) pair represents a specific spot on this plane. The x-value tells us how far to move left or right from the origin (the point where the axes cross), and the y-value tells us how far to move up or down.

Now, let’s take each pair from our table and translate it onto the graph:

• (-3, 4): Start at the origin, move 3 units to the left along the x-axis, and then 4 units up along the y-axis. Mark that spot! This is our first star on the canvas.
• (-2, -1): From the origin, go 2 units left and 1 unit down. Mark it!
• (-1, -4): 1 unit left, 4 units down. Mark!
• (-0.5, -4.75): Half a unit left, almost 5 units down. This one's a bit trickier to place precisely, but do your best! Mark!
• (0, -5): This is our y-intercept! Stay at the origin on the x-axis (since x is 0) and go 5 units down. Mark!
• (0.5, -4.75): Half a unit right, almost 5 units down. Notice how this is a mirror image of the point (-0.5, -4.75).
• (1, -4): 1 unit right, 4 units down. Mark!
• (2, -1): 2 units right, 1 unit down. Mark!
• (3, 4): And finally, 3 units right and 4 units up. Mark!

With all our points plotted, we've got a scatter of dots on the coordinate plane. But these aren't just random dots; they're clues, whispering the shape of our function. The next step is to connect them, but not with straight lines! Remember, y = x^2 - 5 is a quadratic function, and quadratic functions have a distinctive curved shape called a parabola. Think of a gentle U-shape or a graceful arch.

So, carefully sketch a smooth curve that passes through all the points. Don't force it into sharp corners; let it flow naturally. This smooth curve, my friends, is the graph of y = x^2 - 5 over the interval $-3 ext{ less than or equal to } x ext{ less than or equal to } 3$. We’ve visually represented the function! How cool is that?

The act of plotting points and drawing the curve is crucial in understanding functions. It’s not just about memorizing equations; it’s about seeing the relationship between x and y. When you plot the graph, you’re making the abstract concrete, turning an equation into a picture. This visual representation can help you understand key features of the function, like its minimum point (the bottom of the U), its symmetry, and how it changes as x changes. So, guys, embrace the graph! It's your function's story told in a visual form.

Step 3 Analyze the Graph of the Function

We've crunched the numbers, filled the table, and plotted the points, and now, we've got a beautiful parabola staring back at us. But our journey doesn't end here! The graph isn't just a pretty picture; it's a treasure map, holding valuable information about the function y = x^2 - 5. This is where we put on our detective hats and start analyzing what the graph is telling us. Understanding the graph is the key to truly grasping the function's behavior.

First, let's talk about the shape. As we mentioned earlier, the graph is a parabola, that classic U-shaped curve that's the signature of quadratic functions. This shape tells us a lot about how the function behaves. It's symmetrical, meaning if you drew a line down the middle, the two halves would mirror each other. This line of symmetry is a crucial feature of parabolas.

Now, let's find that line of symmetry. It runs right through the vertex, which is the lowest (or highest) point on the parabola. In our case, the vertex is the lowest point, and it sits right at (0, -5). This means our line of symmetry is the vertical line x = 0 (the y-axis). See how the two halves of the parabola perfectly reflect each other across this line?

The vertex itself is a goldmine of information. It's the minimum point of our function. That is, the lowest y-value the function ever reaches is -5, and it happens when x is 0. This is an important characteristic of the function, and it's immediately visible on the graph.

Next, let's look at the intercepts – the points where the graph crosses the x and y axes. We already know the y-intercept: it's (0, -5), which is also our vertex. But what about the x-intercepts? These are the points where the graph crosses the x-axis, meaning y is 0. To find them, we'd need to solve the equation 0 = x^2 - 5. This gives us x = ±√5, which are approximately ±2.24. So, our graph crosses the x-axis at roughly (-2.24, 0) and (2.24, 0). These points tell us where the function's output is zero.

Another thing we can glean from the graph is the function's behavior as x gets larger (both positively and negatively). Notice that as you move away from the vertex along the parabola, the y-values increase. This means that as x gets bigger (in either the positive or negative direction), the function's output also gets bigger. This is a characteristic of parabolas that open upwards (like ours does).

Finally, let's remember our interval: $-3 ext{ less than or equal to } x ext{ less than or equal to } 3$. We only graphed the function within this range of x-values. If we were to extend the graph beyond this interval, the parabola would continue to curve upwards. But for our problem, we're only interested in the part of the graph within these bounds. The true understanding of a graph comes from not just plotting it but also analyzing it. Look for the shape, the vertex, the intercepts, the symmetry, and how the function behaves over its domain. The graph is a visual story, and once you learn to read it, you'll unlock a deeper understanding of functions.

So, there you have it! We've successfully graphed the function y = x^2 - 5 over the interval $-3 ext{ less than or equal to } x ext{ less than or equal to } 3$. Remember, graphing functions is a crucial skill in algebra and beyond. It allows us to visualize mathematical relationships and gain a deeper understanding of how functions work. Keep practicing, and you'll become a graph-plotting pro in no time!