Graphing F(x) = X² - 2x - 8 Finding Vertex, Intercepts, Domain, And Range

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Hey guys! Today, we're diving deep into the world of quadratic functions, and we're going to dissect the function f(x) = x² - 2x - 8. We'll graph it, pinpoint its key features, and understand its behavior. Think of this as your ultimate guide to understanding this particular quadratic function. So, buckle up and let's get started!

Understanding the Basics of Quadratic Functions

Before we jump into the specifics of f(x) = x² - 2x - 8, let's refresh our understanding of quadratic functions in general. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (usually x) is 2. The standard form of a quadratic function is:

f(x) = ax² + bx + c

where a, b, and c are constants, and a is not equal to 0 (otherwise, it would be a linear function). The graph of a quadratic function is a parabola, a U-shaped curve. The parabola can open upwards (if a > 0) or downwards (if a < 0). This 'a' value is super important as it tells us a lot about the shape and direction of our parabola!

Key features of a parabola that we'll be focusing on today include:

  • Vertex: The highest or lowest point on the parabola. It's the turning point of the graph.
  • Axis of Symmetry: A vertical line that divides the parabola into two symmetrical halves. It passes through the vertex.
  • X-intercepts: The points where the parabola intersects the x-axis (where f(x) = 0).
  • Y-intercept: The point where the parabola intersects the y-axis (where x = 0).
  • Domain: The set of all possible input values (x-values) for the function.
  • Range: The set of all possible output values (f(x)-values) for the function.
  • Increasing and Decreasing Intervals: The intervals along the x-axis where the function's values are increasing or decreasing.

Now that we have a solid foundation, let's apply these concepts to our specific function: f(x) = x² - 2x - 8.

Analyzing f(x) = x² - 2x - 8

Let's break down our function f(x) = x² - 2x - 8. Comparing it to the standard form f(x) = ax² + bx + c, we can identify the coefficients:

  • a = 1
  • b = -2
  • c = -8

Since a = 1, which is greater than 0, we know that the parabola opens upwards. This means it will have a minimum point (the vertex) and will look like a smiley face! This is our first key insight into the function's behavior.

Finding the Vertex

The vertex is arguably the most important feature of a parabola. It's the turning point, the minimum or maximum value of the function. There are a couple of ways to find the vertex. One way is by using the vertex formula. The x-coordinate of the vertex, often denoted as h, is given by:

h = -b / 2a

In our case, a = 1 and b = -2, so:

h = -(-2) / (2 * 1) = 2 / 2 = 1

So, the x-coordinate of the vertex is 1. To find the y-coordinate, often denoted as k, we substitute h = 1 back into the original function:

k = f(1) = (1)² - 2(1) - 8 = 1 - 2 - 8 = -9

Therefore, the vertex of the parabola is (1, -9). This is the lowest point on our graph. We've nailed down a crucial point!

Another method to find the vertex involves completing the square. This technique rewrites the quadratic function in vertex form, which directly reveals the vertex coordinates. Let's go through it:

  1. Start with f(x) = x² - 2x - 8
  2. Focus on the first two terms: x² - 2x. To complete the square, we need to add and subtract (b/2)² which in our case is (-2/2)² = 1.
  3. Rewrite the function: f(x) = (x² - 2x + 1) - 1 - 8
  4. Factor the perfect square trinomial: f(x) = (x - 1)² - 9

Now the function is in vertex form: f(x) = a(x - h)² + k, where (h, k) is the vertex. Comparing this to our result, we see that the vertex is indeed (1, -9), confirming our previous calculation. Completing the square not only helps find the vertex but also provides a different perspective on the function's structure.

Determining the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. It's like a mirror running down the middle of the U-shape. The equation of the axis of symmetry is simply:

x = h

where h is the x-coordinate of the vertex. Since we found the vertex to be (1, -9), the axis of symmetry is x = 1. This line is a crucial reference point when graphing the parabola, ensuring that both sides are mirror images of each other.

Finding the X-Intercepts

The x-intercepts are the points where the parabola crosses the x-axis. At these points, the y-value (or f(x) value) is zero. So, to find the x-intercepts, we need to solve the equation:

f(x) = x² - 2x - 8 = 0

This is a quadratic equation, and we can solve it by factoring, using the quadratic formula, or completing the square (again!). Let's try factoring first. We need to find two numbers that multiply to -8 and add up to -2. Those numbers are -4 and 2. So, we can factor the quadratic as:

(x - 4)(x + 2) = 0

Now, set each factor equal to zero and solve for x:

x - 4 = 0 => x = 4 x + 2 = 0 => x = -2

Therefore, the x-intercepts are (4, 0) and (-2, 0). These are the points where the parabola intersects the x-axis. If factoring didn't work, we could always resort to the quadratic formula, which guarantees a solution for any quadratic equation.

Identifying the Y-Intercept

The y-intercept is the point where the parabola crosses the y-axis. At this point, the x-value is zero. To find the y-intercept, we substitute x = 0 into the function:

f(0) = (0)² - 2(0) - 8 = -8

So, the y-intercept is (0, -8). This is the point where the parabola intersects the y-axis. It's a straightforward calculation, making it one of the easier key features to find.

Defining the Domain and Range

The domain of a quadratic function is always all real numbers. This means that we can input any value for x and get a valid output. In interval notation, the domain is (-∞, ∞). Parabolas stretch infinitely to the left and right, covering all possible x-values.

The range, on the other hand, is restricted by the vertex. Since our parabola opens upwards and the vertex is the minimum point, the range includes all y-values greater than or equal to the y-coordinate of the vertex. The y-coordinate of the vertex is -9, so the range is:

Range: [-9, ∞)

This means the function's output values will always be -9 or higher. The square bracket indicates that -9 is included in the range, as it's the minimum value.

Determining Increasing and Decreasing Intervals

A function is increasing when its y-values are increasing as we move from left to right along the x-axis. Conversely, it's decreasing when its y-values are decreasing as we move from left to right. For a parabola that opens upwards, like ours, the function decreases to the left of the vertex and increases to the right of the vertex.

The vertex is at x = 1. So:

  • The function is decreasing on the interval (-∞, 1).
  • The function is increasing on the interval (1, ∞).

These intervals tell us how the function's values change as x increases. The behavior is symmetric around the axis of symmetry.

Graphing f(x) = x² - 2x - 8

Now that we've found all the key features, let's put it all together and graph the function f(x) = x² - 2x - 8. We have:

  • Vertex: (1, -9)
  • Axis of symmetry: x = 1
  • X-intercepts: (4, 0) and (-2, 0)
  • Y-intercept: (0, -8)
  1. Plot the Vertex: Start by plotting the vertex (1, -9) on the coordinate plane. This is the lowest point of our parabola.
  2. Draw the Axis of Symmetry: Draw a vertical dashed line at x = 1. This line will help us create a symmetrical graph.
  3. Plot the Intercepts: Plot the x-intercepts (4, 0) and (-2, 0) and the y-intercept (0, -8).
  4. Sketch the Parabola: Now, carefully sketch the parabola, making sure it passes through the intercepts and has the correct U-shape. Remember, the parabola is symmetrical about the axis of symmetry. The curve should be smooth and extend upwards from the vertex.

By plotting these key points and using the axis of symmetry as a guide, you can accurately graph the quadratic function. The graph visually represents all the information we've calculated: the vertex as the minimum point, the intercepts as the points where the parabola crosses the axes, and the overall U-shape.

Summary of Key Features

To recap, let's summarize the key features of f(x) = x² - 2x - 8:

  • Vertex: (1, -9)
  • Axis of Symmetry: x = 1
  • X-intercepts: (4, 0) and (-2, 0)
  • Y-intercept: (0, -8)
  • Domain: (-∞, ∞)
  • Range: [-9, ∞)
  • Increasing Interval: (1, ∞)
  • Decreasing Interval: (-∞, 1)

We've thoroughly analyzed this quadratic function, from finding its vertex and intercepts to determining its domain, range, and increasing/decreasing intervals. Understanding these features allows us to accurately graph the function and predict its behavior. You guys have now got a complete picture of this quadratic function!

Conclusion

Graphing quadratic functions might seem daunting at first, but by breaking it down into smaller steps and understanding the key features, it becomes much more manageable. We've successfully graphed f(x) = x² - 2x - 8, found its vertex, axis of symmetry, intercepts, domain, range, and increasing/decreasing intervals. Remember, practice makes perfect, so try graphing other quadratic functions to solidify your understanding. You've got this!