Graphing Quadratic Functions: A Step-by-Step Guide

Hey everyone! Today, we're going to dive into the world of quadratic functions. Specifically, we'll be tackling the function f(x) = x² - 8x. Don't worry, it's not as scary as it sounds! We'll break it down step by step and you'll be graphing quadratic functions like a pro in no time. This guide will walk you through finding all the key features you need to sketch the graph accurately. Let's get started!

(a) Graphing the Quadratic Function: Unveiling the Secrets of the Curve

Alright guys, the first thing we need to do is graph the quadratic function f(x) = x² - 8x. To do this, we need a few key pieces of information. This includes figuring out whether the graph opens up or down, identifying the vertex, finding the axis of symmetry, determining the y-intercept, and locating any x-intercepts. Understanding these elements is crucial for accurately sketching the curve. Let's start with the basics.

First things first: does the graph open up or down? This is super easy to figure out. Look at the coefficient of the x² term. In our case, it's 1 (since we have x²). If the coefficient is positive, the parabola opens upwards (like a smile!). If it's negative, it opens downwards (like a frown!). Since our coefficient is positive (1), our parabola opens upwards. Knowing this immediately gives us an idea of the general shape of the graph.

Next up, we need to find the vertex. The vertex is the most important point on the parabola. It's either the lowest point (if the parabola opens up) or the highest point (if it opens down). The vertex is like the turning point of the graph. We can find the x-coordinate of the vertex using the formula x = -b / 2a, where a and b are the coefficients from our quadratic equation in the form ax² + bx + c. In our function, f(x) = x² - 8x, we have a = 1 and b = -8. Plugging these values into the formula gives us x = -(-8) / (2 * 1) = 8 / 2 = 4. So, the x-coordinate of our vertex is 4. To find the y-coordinate, we plug this x-value back into the original equation: f(4) = (4)² - 8(4) = 16 - 32 = -16. Therefore, the vertex of our parabola is the point (4, -16). Remember that the vertex is an ordered pair (x, y). The vertex is very important, because it gives us a good reference point for the whole graph. The vertex gives the parabola it's turning point.

Now, let's find the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex. It essentially divides the parabola into two symmetrical halves. The equation of the axis of symmetry is always x = the x-coordinate of the vertex. Since our vertex has an x-coordinate of 4, the axis of symmetry is the line x = 4. The axis of symmetry is important, because it tells us that both sides of the parabola are mirrors of each other. The axis of symmetry always goes through the vertex, and provides additional guidance when plotting points.

Moving on to the y-intercept. The y-intercept is the point where the graph crosses the y-axis. This happens when x = 0. So, to find the y-intercept, we substitute x = 0 into our equation: f(0) = (0)² - 8(0) = 0 - 0 = 0. Thus, the y-intercept is the point (0, 0). This point is also the origin in this specific case. It's useful to know the y-intercept because it gives us another point to plot on the graph, providing additional shape information. This also gives a reference point to where the parabola intersects the y axis.

Finally, let's find the x-intercepts, also known as the zeros or roots of the function. The x-intercepts are the points where the graph crosses the x-axis. This happens when f(x) = 0. So, we need to solve the equation x² - 8x = 0. We can solve this by factoring out an x: x(x - 8) = 0. This means either x = 0 or x - 8 = 0, which gives us x = 8. Therefore, the x-intercepts are (0, 0) and (8, 0). The x-intercepts are essential, because they show the points at which the parabola intersects the x-axis. These points are also known as the zeros of the function, and are very important in finding the range of the function, and understanding how the function behaves. These points will assist in understanding when the function turns positive and negative.

With all this information – the direction it opens, the vertex, the axis of symmetry, the y-intercept, and the x-intercepts – we have all the tools needed to sketch the graph of the quadratic function accurately! You can now plot these points and draw a smooth curve to represent the parabola.

(b) Finding the Domain and Range

Alright, now that we've graphed our quadratic function, let's talk about its domain and range. These are important concepts that describe the possible input values (domain) and output values (range) of a function. Let's get into it.

First, let's look at the domain. For any quadratic function, the domain is always all real numbers. This means that we can plug in any real number for x and the function will produce a valid output. Think about it: you can square any number and subtract 8 times that number. There are no restrictions on the values of x we can use. Mathematically, we represent this as (-∞, ∞), which means all numbers from negative infinity to positive infinity. The domain is straightforward for quadratic functions. The main reason is because there are no fractions or square roots in the function.

Next, let's find the range. The range, on the other hand, depends on whether the parabola opens up or down and where the vertex is located. Since our parabola opens upwards (as we determined in part (a)), it has a minimum value but no maximum value. The minimum value is the y-coordinate of the vertex. From part (a), we know our vertex is at (4, -16). So, the minimum value of the function is -16. This means the parabola goes as low as -16, but then it goes upwards infinitely. Therefore, the range of our function is all real numbers greater than or equal to -16. Mathematically, we represent this as [-16, ∞). This indicates that the function includes all y-values from -16 (inclusive) and goes up to positive infinity. To summarize, because the parabola opens upward, the range begins with the y-coordinate of the vertex and goes to infinity. If the parabola opened downward, the range would begin at negative infinity and end with the y-coordinate of the vertex.

Understanding domain and range is important for comprehending the behavior of the function. In simple terms, the domain helps us understand what kind of input values are allowed. Whereas the range helps us understand what kind of output values are generated by the function.

(c) Determine Where f(x) is Increasing or Decreasing

Alright everyone, let's explore the final part of our problem: determining where the function f(x) = x² - 8x is increasing or decreasing. This tells us about the direction of the graph along the x-axis. This is useful for understanding the function's behavior.

Remember how the vertex is the turning point of the parabola? That's key to understanding where the function increases and decreases. The axis of symmetry, x = 4, that we calculated in part (a) is also very important here. This line of symmetry is really the divide between the increasing and decreasing segments of the parabola.

Because our parabola opens upwards, it decreases to the left of the vertex and increases to the right of the vertex. More specifically, the function is decreasing when x is less than the x-coordinate of the vertex, which is 4. Mathematically, this is represented as (-∞, 4). This interval means that as we move along the x-axis from negative infinity up to 4, the value of the function f(x) decreases.

On the other hand, the function is increasing when x is greater than the x-coordinate of the vertex, which is 4. This is represented as (4, ∞). This means that as we move along the x-axis from 4 to positive infinity, the value of the function f(x) increases.

To recap: the function f(x) = x² - 8x is decreasing on the interval (-∞, 4) and increasing on the interval (4, ∞). This information helps us understand the shape and behavior of the parabola and is helpful for many related calculations.

And that's it! We've successfully analyzed the quadratic function f(x) = x² - 8x. We've graphed it, found its domain and range, and determined where it's increasing or decreasing. You guys did great! Keep practicing and you'll become quadratic function experts in no time!