Analyzing The Quadratic Function F(x) = (x+4)(x-6) Vertex And Intervals

This article delves into the analysis of the quadratic function $f(x) = (x+4)(x-6)$, focusing on key features identifiable from its graph. We will explore how to determine the vertex, intervals of increase and decrease, and other significant characteristics. Understanding these aspects is crucial for grasping the behavior of quadratic functions and their applications in various fields.

Determining the Vertex of the Quadratic Function

To accurately determine the vertex of the quadratic function $f(x) = (x+4)(x-6)$, we need to understand what the vertex represents. The vertex is the point where the parabola, which is the graphical representation of a quadratic function, changes direction. It is either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). For a quadratic function in the standard form $f(x) = ax^2 + bx + c$, or in the factored form as given, the vertex's x-coordinate can be found using the formula $x = -b / 2a$ or by finding the midpoint of the roots. Let's first expand the given function to get it into the standard form:

$f(x) = (x+4)(x-6) = x^2 - 6x + 4x - 24 = x^2 - 2x - 24$

Here, $a = 1$, $b = -2$, and $c = -24$. Using the formula for the x-coordinate of the vertex:

$x = -(-2) / (2 * 1) = 2 / 2 = 1$

Now, we substitute $x = 1$ back into the function to find the y-coordinate of the vertex:

$f(1) = (1)^2 - 2(1) - 24 = 1 - 2 - 24 = -25$

Therefore, the vertex of the function is at the point $(1, -25)$. This indicates that the parabola has a minimum point at this location. The negative y-coordinate also tells us that the vertex lies below the x-axis. Understanding the vertex is pivotal, as it helps in visualizing the overall shape and position of the parabola on the coordinate plane. It also aids in solving optimization problems where we need to find the maximum or minimum value of a quadratic function. The vertex form of a quadratic equation, $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex, provides an alternative representation that directly reveals the vertex coordinates. In our case, the vertex form would be $f(x) = (x-1)^2 - 25$, further confirming our calculated vertex. By correctly identifying the vertex, we can accurately describe the behavior of the quadratic function and its graphical representation.

Analyzing the Increasing and Decreasing Intervals of the Function

Analyzing the increasing and decreasing intervals of the function $f(x) = (x+4)(x-6)$ is crucial for understanding its behavior. A function is said to be increasing over an interval if its y-values increase as its x-values increase, and decreasing if its y-values decrease as its x-values increase. For a parabola, the increasing and decreasing intervals are separated by the vertex. Since we've already determined that the vertex of the function is at $(1, -25)$, we can use this information to find the intervals.

As the coefficient of the $x^2$ term in the expanded form ($f(x) = x^2 - 2x - 24$) is positive ($a = 1$), the parabola opens upwards. This means that the function decreases to the left of the vertex and increases to the right of the vertex. The vertex represents the minimum point of the function.

Therefore, the function is decreasing on the interval $(-\infty, 1)$. This means that as x-values move from negative infinity towards 1, the y-values of the function decrease. Graphically, this corresponds to the left side of the parabola sloping downwards. Once the graph reaches the vertex at $x = 1$, it changes direction and starts to increase.

Conversely, the function is increasing on the interval $(1, \infty)$. This means that as x-values move from 1 towards positive infinity, the y-values of the function increase. Graphically, this corresponds to the right side of the parabola sloping upwards. Understanding these intervals is vital for applications such as optimization problems, where we want to find the minimum or maximum value of a function. The increasing and decreasing behavior helps us to pinpoint where these extreme values occur. Furthermore, this analysis provides a comprehensive picture of how the function behaves across its domain, allowing for accurate predictions and interpretations.

To summarize, the function $f(x) = (x+4)(x-6)$ decreases from $(-\infty)$ to $x=1$ and increases from $x=1$ to $(\infty)$. This behavior is directly tied to the parabolic shape of the function and the location of its vertex.

Roots and Intercepts of the Quadratic Function

Understanding the roots and intercepts of a quadratic function is fundamental to analyzing its graph and behavior. The roots, also known as the x-intercepts, are the points where the graph of the function intersects the x-axis. These are the values of $x$ for which $f(x) = 0$. The y-intercept, on the other hand, is the point where the graph intersects the y-axis, which occurs when $x = 0$.

For the function $f(x) = (x+4)(x-6)$, the roots are easily found by setting the function equal to zero and solving for $x$:

$(x+4)(x-6) = 0$

This equation is satisfied when either $(x+4) = 0$ or $(x-6) = 0$. Solving these equations gives us the roots:

$x+4 = 0 \Rightarrow x = -4$ $x-6 = 0 \Rightarrow x = 6$

Thus, the roots of the function are $x = -4$ and $x = 6$. These points, $(-4, 0)$ and $(6, 0)$, are where the parabola intersects the x-axis. The roots provide crucial information about the parabola's position and symmetry. The axis of symmetry, which is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves, lies exactly in the middle of the roots. In this case, the axis of symmetry is the vertical line $x = 1$, which we already found to be the x-coordinate of the vertex.

To find the y-intercept, we set $x = 0$ in the function:

$f(0) = (0+4)(0-6) = 4 * (-6) = -24$

So, the y-intercept is at the point $(0, -24)$. This is the point where the parabola intersects the y-axis. The y-intercept, along with the roots and the vertex, gives us a clear picture of the parabola's orientation and position on the coordinate plane. In summary, the roots of the function $f(x) = (x+4)(x-6)$ are $x = -4$ and $x = 6$, and the y-intercept is at $(0, -24)$. These key features, combined with the vertex and the increasing/decreasing intervals, provide a comprehensive understanding of the quadratic function's behavior.

Conclusion

In conclusion, the analysis of the quadratic function $f(x) = (x+4)(x-6)$ reveals several key features. The vertex, calculated to be at $(1, -25)$, represents the minimum point of the parabola and the turning point of the function. The function decreases on the interval $(-\infty, 1)$ and increases on the interval $(1, \infty)$. The roots, found at $x = -4$ and $x = 6$, are the points where the parabola intersects the x-axis, and the y-intercept is at $(0, -24)$. These elements together provide a complete understanding of the function's graph and behavior. By understanding these concepts, one can effectively analyze and interpret quadratic functions in various mathematical and real-world contexts.