Graphing F(x) = (x+1)(x-5): Vertex & Intercept Method

by TextBrain Team 54 views

Alright, let's dive into graphing the quadratic function f(x) = (x+1)(x-5). We're going to focus on plotting the vertex and one of the x-intercepts to get a good idea of what this parabola looks like. This method is super handy because it uses key features of the function that are relatively easy to find.

1. Finding the X-Intercepts

First things first, let's find those x-intercepts. Remember, x-intercepts are the points where the graph crosses the x-axis, meaning f(x) = 0. So, we need to solve the equation (x+1)(x-5) = 0. This is already factored for us, which is awesome!

The product of two factors is zero if and only if at least one of the factors is zero. This gives us two possible solutions:

  • x + 1 = 0 which implies x = -1
  • x - 5 = 0 which implies x = 5

So, our x-intercepts are (-1, 0) and (5, 0). We've already got two key points to plot! Let's hold onto these for a moment while we find the vertex.

2. Locating the Vertex

The vertex is the highest or lowest point on the parabola. Since our parabola opens upwards (the coefficient of the x^2 term is positive after expanding), the vertex will be the minimum point. There are a couple of ways to find the vertex, but since we already have the x-intercepts, let's use that information.

The x-coordinate of the vertex is exactly halfway between the two x-intercepts. This is because parabolas are symmetrical. So, we can find the x-coordinate of the vertex by averaging the x-intercepts:

  • x_vertex = (-1 + 5) / 2 = 4 / 2 = 2

Now that we have the x-coordinate of the vertex (x = 2), we can find the y-coordinate by plugging it back into our original function:

  • f(2) = (2 + 1)(2 - 5) = (3)(-3) = -9

Therefore, the vertex of our parabola is (2, -9). This is the lowest point on our graph.

3. Plotting the Points and Sketching the Graph

Now comes the fun part: plotting! Grab your graph paper (or your favorite graphing software) and let's plot the points we've found:

  1. Plot the x-intercepts: (-1, 0) and (5, 0).
  2. Plot the vertex: (2, -9).

Now, sketch a smooth curve that passes through these three points. Remember that a parabola is symmetrical, so the curve should look the same on both sides of the vertex. Since the coefficient of x^2 is positive when we expand the original equation, the parabola opens upward.

Additional Tips for Graphing

  • Symmetry: Use the symmetry of the parabola to your advantage. If you want to plot another point, choose an x-value, find its corresponding y-value, and then use symmetry to find another point on the other side of the vertex.
  • The y-intercept: Find the y-intercept by setting x = 0 in the original equation: f(0) = (0 + 1)(0 - 5) = -5. So the y-intercept is (0, -5). This gives you another point to help guide your sketch.
  • Check your work: Use a graphing calculator or online graphing tool (like Desmos or GeoGebra) to check your graph. This is a great way to make sure you haven't made any mistakes.

4. Expanding the Function (Optional, but Useful)

While we didn't need to expand the function to find the vertex and x-intercepts, it can be helpful to see the function in standard quadratic form, which is f(x) = ax^2 + bx + c. Let's expand our function:

f(x) = (x + 1)(x - 5) f(x) = x(x - 5) + 1(x - 5) f(x) = x^2 - 5x + x - 5 f(x) = x^2 - 4x - 5

From this form, we can see that a = 1, b = -4, and c = -5. The coefficient 'a' tells us whether the parabola opens upwards (a > 0) or downwards (a < 0). In this case, since a = 1, the parabola opens upwards, which confirms what we already knew.

Also, the y-intercept is simply the value of 'c', which is -5. So the y-intercept is (0, -5), which matches what we found earlier.

Using the Quadratic Formula (Alternative Method for X-Intercepts)

If the function wasn't already factored, we could have used the quadratic formula to find the x-intercepts. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 1, b = -4, and c = -5. Plugging these values into the quadratic formula gives us:

x = (4 ± √((-4)^2 - 4 * 1 * -5)) / (2 * 1) x = (4 ± √(16 + 20)) / 2 x = (4 ± √36) / 2 x = (4 ± 6) / 2

This gives us two solutions:

  • x = (4 + 6) / 2 = 10 / 2 = 5
  • x = (4 - 6) / 2 = -2 / 2 = -1

These are the same x-intercepts we found earlier: x = -1 and x = 5. So, the quadratic formula is a reliable way to find the x-intercepts, even if the function isn't factored.

5. Wrapping It Up

So, to recap, graphing f(x) = (x+1)(x-5) by plotting the vertex and one of the x-intercepts involves these steps:

  1. Find the x-intercepts: Set f(x) = 0 and solve for x. This gives you the points where the graph crosses the x-axis.
  2. Find the vertex: The x-coordinate of the vertex is the average of the x-intercepts. Plug this x-value back into the original function to find the y-coordinate of the vertex.
  3. Plot the points: Plot the x-intercepts and the vertex on a coordinate plane.
  4. Sketch the graph: Draw a smooth, symmetrical curve through the points, remembering that the parabola opens upwards if the coefficient of x^2 is positive, and downwards if it's negative.

By following these steps, you can quickly and easily graph quadratic functions by hand. Remember to use additional points and the symmetry of the parabola to make your graph more accurate. And don't forget to check your work with a graphing calculator or online tool!

Keep practicing, guys, and you'll become graphing pros in no time!