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Chapter 9: Problem 13

Identify the vertex of each parabola. $$ f(x)=(x+3)^{2}-4 $$

Short Answer

Expert verified
The vertex is at (-3, -4).

Step by step solution

01

Identify the Standard Form

Recognize that the given equation is in the vertex form, which is generally written as \[ f(x) = a(x-h)^2 + k \]. Identify the values of \( h \) and \( k \) by comparing it to \( f(x) = (x+3)^2-4 \).
02

Extract Vertex Coordinates

From the equation \( f(x) = (x+3)^2-4 \), recognize that it matches the form \( (x-h)^2+k \). Therefore, \( h = -3 \) and \( k = -4 \).
03

State the Vertex

The vertex of the parabola is at the point \( (h, k) \), which is \( (-3, -4) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of a Parabola
The standard form of a parabola is an important concept in algebra and usually comes in the shape of a quadratic equation:

\[ y = ax^2 + bx + c \]

In this equation,
  • \(a\) controls the direction and width of the parabola. When \(a\) is positive, the parabola opens upwards. When \(a\) is negative, it opens downwards.
  • \(b\) is the coefficient that affects the placement of the vertex along the x-axis.
  • \(c\) is the y-intercept, the point where the parabola crosses the y-axis.
Recognizing how these coefficients interact is crucial for sketching the graph of the parabola and for understanding its properties.
Vertex Form
The vertex form of a parabola makes it easy to identify the vertex. The formula for the vertex form is:

\[ f(x) = a(x-h)^2 + k \]

Here,
  • \(a\) still affects the direction and width of the parabola.
  • \(h\) is the x-coordinate of the vertex. The sign in the equation \( (x-h) \) tells you the direction: if it says \((x-h)\) then \(h\) is positive; if it says \((x+3)\) as in our exercise, then \(h\) is actually \(-3\).
  • \(k\) is the y-coordinate of the vertex.
Using the vertex form, you can quickly determine the highest or lowest point on the graph, which is the vertex. This form is especially helpful in solving problems that require you to identify the vertex, just like in our exercise.
Coordinate Calculation
Identifying the vertex from the vertex form of a parabola is a straightforward process of coordinate calculation:

#### Identifying the Vertex CoordinatesTo find the vertex, simply look at the vertex form equation

f(x) = (x+3)^2 - 4.
  • Here, \( (x+3) \) means \( h = -3 \)
  • while \( -4 \) means \( k = -4 \).

    So, the vertex \( (h, k) \) would be \ (-3, -4) \.

    • Remember, to identify \( h \) correctly, change the sign inside the parenthesis.
    • For \( k \), simply use the constant term outside the parenthesis.
    Given the equation in vertex form makes finding the vertex a breeze - you just

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