Analyzing F(x) = (x-1)(x+7): Vertex And Graph Properties

Hey guys! Let's dive into understanding the function f(x) = (x-1)(x+7) and explore its graph. We're going to pinpoint the vertex and discuss other key characteristics. This is super important in mathematics, especially when you're dealing with quadratic functions. So, grab your thinking caps, and let’s get started!

Understanding the Quadratic Function

First off, let's recognize that f(x) = (x-1)(x+7) is a quadratic function. You can tell because when you expand it, you get a polynomial of degree two. Expanding it gives us:

f(x) = x^2 + 7x - x - 7 = x^2 + 6x - 7

This is the standard form of a quadratic function, which is f(x) = ax^2 + bx + c, where a, b, and c are constants. In our case, a = 1, b = 6, and c = -7. Understanding these coefficients is crucial because they tell us a lot about the parabola's shape and position. For example, since 'a' is positive (a = 1), we know that the parabola opens upwards, meaning it has a minimum point, which is the vertex.

Finding the Vertex

The vertex is a critical point on the parabola. It’s the point where the parabola changes direction. For a parabola that opens upwards, like ours, the vertex is the lowest point. There are a couple of ways to find the vertex. One common method involves using the formula for the x-coordinate of the vertex, which is given by:

x_vertex = -b / (2a)

In our function, b = 6 and a = 1, so:

x_vertex = -6 / (2 * 1) = -3

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging x = -3 back into our function:

f(-3) = (-3)^2 + 6*(-3) - 7 = 9 - 18 - 7 = -16

So, the vertex of the function is at (-3, -16). This means the statement “The vertex of the function is at (-3, -16)” is correct! Isn't it cool how the formula helps us find such an important point on the graph?

Knowing the vertex is super helpful because it gives us a central reference point for the parabola. We know that the parabola is symmetric around the vertical line that passes through the vertex. This line is called the axis of symmetry, and its equation is x = -3 in our case.

Analyzing the Graph

Let’s think about what the graph of f(x) = (x-1)(x+7) looks like. We already know it’s a parabola opening upwards with a vertex at (-3, -16). But there's more we can figure out. The factored form of the function, (x-1)(x+7), gives us the x-intercepts, which are the points where the parabola crosses the x-axis. These occur when f(x) = 0. So, we set (x-1)(x+7) = 0, which gives us x = 1 and x = -7.

These x-intercepts (1, 0) and (-7, 0) give us two more key points on the graph. They also help confirm our vertex calculation. Remember, the vertex lies on the axis of symmetry, which is exactly in the middle of the x-intercepts. The midpoint between x = -7 and x = 1 is:

((-7) + 1) / 2 = -6 / 2 = -3

This matches our calculated x-coordinate of the vertex, which is a good sign that we're on the right track!

Now, imagine sketching this parabola. It passes through (-7, 0), (1, 0), and has its lowest point at (-3, -16). It's symmetrical around the line x = -3, and it opens upwards. Visualizing the graph like this can help you answer a variety of questions about the function's behavior.

Additional Statements and Function Properties

We've confirmed the vertex, but what about other properties of the function? Let's dive deeper into analyzing additional statements and understanding various aspects of quadratic functions.

Exploring the Axis of Symmetry

As we discussed earlier, the axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. For our function f(x) = (x-1)(x+7), the axis of symmetry is the line x = -3. This means that if you were to fold the graph along this line, the two halves would match perfectly. This symmetry is a fundamental characteristic of parabolas, and understanding it can help you sketch the graph and identify key points more easily.

The axis of symmetry also helps us understand the function's behavior. For example, any point on the parabola has a corresponding point on the other side of the axis of symmetry that has the same y-value. This property can be useful for finding additional points on the graph or for solving equations involving quadratic functions.

Analyzing the Domain and Range

Another important aspect of any function is its domain and range. The domain is the set of all possible input values (x-values) for which the function is defined, and the range is the set of all possible output values (y-values). For a quadratic function like f(x) = (x-1)(x+7), the domain is all real numbers because you can plug in any x-value and get a valid output. There are no restrictions, such as division by zero or taking the square root of a negative number.

However, the range is limited because the parabola has a minimum point at the vertex. Since the vertex is at (-3, -16) and the parabola opens upwards, the y-values will always be greater than or equal to -16. Therefore, the range of the function is y ≥ -16. Understanding the range is crucial for applications of quadratic functions, such as optimization problems where you want to find the minimum or maximum value of a quantity.

Determining Intervals of Increase and Decrease

Quadratic functions also have intervals where they are increasing or decreasing. A function is said to be increasing if its y-values increase as x-values increase, and it is decreasing if its y-values decrease as x-values increase. For our parabola, which opens upwards, the function decreases to the left of the vertex and increases to the right of the vertex.

Specifically, f(x) is decreasing on the interval (-∞, -3) and increasing on the interval (-3, ∞). The vertex at x = -3 is the point where the function transitions from decreasing to increasing. Knowing these intervals can help you sketch the graph accurately and predict the function's behavior over different ranges of x-values.

Identifying Key Points: Intercepts and More

Besides the vertex and the x-intercepts, identifying other key points can provide a comprehensive understanding of the function's graph. The y-intercept, where the parabola crosses the y-axis, is one such point. To find the y-intercept, we set x = 0 in the function:

f(0) = (0)^2 + 6*(0) - 7 = -7

So, the y-intercept is (0, -7). This gives us another point to plot on the graph and helps us visualize the curve of the parabola.

Additionally, you can choose other arbitrary x-values and calculate the corresponding y-values to plot more points. This can be particularly useful if you need to sketch the graph by hand or if you want to verify the accuracy of your calculations. For example, you might choose x = -1 or x = -5 to find additional points on the parabola.

Conclusion

Alright, guys, we've taken a deep dive into the function f(x) = (x-1)(x+7) and its graph. We figured out the vertex is at (-3, -16), which is super important. We also talked about the x-intercepts, the axis of symmetry, the domain and range, and the intervals where the function increases or decreases. By understanding these properties, you can confidently analyze and sketch quadratic functions. Keep practicing, and you'll become a pro at graphing parabolas in no time! Remember, math is like a puzzle, and each piece we learn helps us see the bigger picture. Keep up the awesome work!"