Problem 44 - Introductory Algebra for College Students

Question

Find the vertex for the parabola whose equation is given by first writing the equation in the form \(y=a x^{2}+b x+c\) $$y=-3(x+4)^{2}+6$$

Step-by-Step Solution

Answer

The vertex of the parabola given by the equation \(y=-3(x+4)^2 + 6\) is (-4, 6).

Step 1: Identify the standard form

The standard form of a parabola is \(y=a(x-h)^2 + k\), where (h, k) is the vertex. In this equation, a is the coefficient of \(x^2\), h is the value being subtracted from x, and k is the constant. In this case, the given equation of the parabola is \(y=-3(x+4)^2 + 6\).

Step 2: Identify the vertex

The vertex (h,k) can be found by checking the standard form. In this case, as the form is \(y=-3(x+4)^2 + 6\), h is -4 and k is 6. However, it's important to note that the h value is the opposite value of what is inside the parentheses. So, because it says \(x+4\), h is -4. The k value is the same as the constant, 6.

Step 3: Write the vertex

In light of the analysis, the vertex is (-4, 6).

Key Concepts

Understanding the Standard Form of a Parabola

The standard form of a parabola is a way of expressing a quadratic function that makes it easier to identify certain features of the parabola, such as its vertex. The equation is written as \[y = ax^2 + bx + c\]

• In this equation, \(a\), \(b\), and \(c\) are constants.
• The coefficient \(a\) will affect the parabola's direction and how "wide" or "narrow" it appears.

It's essential to understand that in this form, it might not be immediately obvious what the vertex of the parabola is. To find the vertex here, one usually needs to complete the square or convert to what is known as the vertex form.

Exploring the Vertex Form of a Parabola

The vertex form of a parabola provides a direct and clear way to determine the vertex by structuring the equation as:\[y = a(x-h)^2 + k\]

• Here, \((h, k)\) represents the vertex of the parabola.
• This form makes it intuitive to understand that modifying \(h\) and \(k\) shifts the parabola left/right and up/down.

If an equation is presented in a different form, such as the standard form, it can be converted into the vertex form. In the exercise above, the given equation \(y = -3(x+4)^2 + 6\) is already in vertex form, making it easy to identify the vertex as \((-4, 6)\). The expression \((x+4)\) means \(h\) is \(-4\), as \(h\) is the value that makes the expression zero, the opposite of what's inside the parentheses.

Delving into the Parabola Equation

A parabola equation explains the set of all points that form a symmetrical curve on a plane. The most common forms of this equation are the standard form and the vertex form. Both forms express quadratic relationships:

• The standard form: \(y = ax^2 + bx + c\)
• The vertex form: \(y = a(x-h)^2 + k\)

Both equations describe how the curve appears in a Cartesian plane, but they serve different purposes. For instance, the vertex form easily reveals the vertex, offering a straightforward way to graph the parabola based on its shape and position. The constant \(a\) in both forms shapes the parabola's width and direction. A positive \(a\) value makes the parabola open upwards, whereas a negative \(a\) flips it to open downwards. By understanding these equations, one can accurately work with and graph parabolas.