Problem 23 - Algebra 2

Question

Identify the vertex and the \(y\) -intercept of the graph of each function. $$ y=24(x+5.5)^{2} $$

Step-by-Step Solution

Answer

The vertex of the function is at the point (-5.5, 0), and the y-intercept is at the point (0, 24).

Step 1: Identify the vertex of the quadratic function

Given the function in vertex form, which is written as for some value of , , and . In this case, and . The vertex of the function is at the point ().

Step 2: Identify the y-intercept of the quadratic function

The y-intercept occurs when . Substitute into the equation to find the y-intercept. Therefore, the y-intercept is at the point ().

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is critical when analyzing quadratic functions. The vertex is the highest or lowest point on the parabola, which is a U-shaped curve. In a quadratic function with standard form as \(y = ax^2 + bx + c\), finding the vertex requires either completing the square or differentiating the function if you're familiar with calculus.

However, for a function in vertex form, \(y = a(x-h)^2 + k\), the task is much simpler. The vertex can be identified directly from the equation as the point \((h, k)\). In the given exercise, our function is \(y = 24(x+5.5)^2\), we can see that \(h = -5.5\) and because there is no \(k\) value, it implies \(k = 0\). Therefore, the vertex is at the point \((-5.5, 0)\), representing the lowest point of the parabola since the coefficient of the squared term is positive.

Y-Intercept

The point where the graph of a function crosses the y-axis is known as the y-intercept. For any function, the y-intercept can be determined by evaluating the function when \(x = 0\).

In the context of quadratic functions, finding the y-intercept allows us to understand one specific point the parabola will pass through on the coordinate plane. For our function \(y = 24(x+5.5)^2\), we set \(x=0\) and calculate the resulting y-value to determine the y-intercept. Plugging in \(x = 0\) gives us \(y = 24(0+5.5)^2\) or \(y = 24(5.5)^2\), which simplifies to the y-intercept at the point \((0, 726)\). This information is particularly useful when graphing the quadratic function.

Vertex Form of a Quadratic Equation

A quadratic equation can be expressed in different forms, and one of the most instructive ones is the vertex form. The vertex form of a quadratic equation is \(y = a(x-h)^2 + k\), where \(a\) represents the dilation and direction of the parabola, while \(h\) and \(k\) represent the coordinates of the vertex, as mentioned earlier.

The primary advantage of this form is its direct revelation of the vertex and the way it simplifies the process of graphing the function. When you want to identify the vertex quickly or when graphing by hand, converting a quadratic equation to vertex form can be remarkably helpful. To convert a standard form equation to vertex form, you can use the process of completing the square or work backwards from the given information if you're starting with a graph or certain points.

Graphing Quadratic Functions

Graphing quadratic functions is a foundational skill in algebra. Each quadratic function graphs into a parabola that can open either upward or downward. To graph a quadratic function, you first need to find key features like the vertex, y-intercept, and possibly the x-intercepts.

Plotting the vertex and y-intercept sets up a