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Chapter 9: Problem 10

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

Short Answer

Expert verified
The vertex of the parabola is \((0, -4)\).

Step by step solution

01

- Understand the given quadratic function

The given function is in the form of a quadratic equation, which is typically represented as \( f(x) = ax^2 + bx + c \). For this exercise, the function is: \( f(x) = x^2 - 4 \). Here, \( a = 1 \), \( b = 0 \), and \( c = -4 \).
02

- Recall the vertex formula for a parabola

The vertex of a parabola given by the quadratic function \( f(x) = ax^2 + bx + c \) can be found using the formula: \( x = -\frac{b}{2a} \). This formula helps find the x-coordinate of the vertex.
03

- Apply the vertex formula

Substitute the values of \( a \) and \( b \) from the given function into the vertex formula: \( x = -\frac{0}{2(1)} = 0 \). Therefore, the x-coordinate of the vertex is \(0\).
04

- Find the y-coordinate of the vertex

Substitute \( x = 0 \) back into the original function to find the y-coordinate: \( f(0) = (0)^2 - 4 = -4 \). Therefore, the y-coordinate of the vertex is \(-4\).
05

- State the vertex

Combine the x-coordinate and y-coordinate to state the vertex of the parabola: The vertex is \((0, -4)\).

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Key Concepts

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

quadratic function
A quadratic function is a polynomial equation of degree 2. It can be represented in the general form: \[ f(x) = ax^2 + bx + c \]
Where:
  • \( a \) is the coefficient of the square term \( x^2 \)
  • \( b \) is the coefficient of the linear term \( x \)
  • \( c \) is the constant term

In the given exercise, our function is \( f(x) = x^2 - 4 \). Here,
  • \( a = 1 \)
  • \( b = 0 \)
  • \( c = -4 \)

Quadratic functions form a graph called a parabola. The parabola could open upwards if \( a > 0 \) (like a U-shape), or downwards if \( a < 0 \) (like an inverted U-shape).
vertex formula
The vertex of a parabola is a point where the curve changes direction. For the quadratic function \( f(x) = ax^2 + bx + c \), the vertex can be found using the vertex formula:
\[ x = -\frac{b}{2a} \]
Let's apply this to our given function \( f(x) = x^2 - 4 \). We know:
  • \( a = 1 \)
  • \( b = 0 \)

Substituting these values into the formula:
\[ x = -\frac{0}{2 \times 1} = 0 \]
So, the x-coordinate of the vertex is \( 0 \).
Next, we need to find the y-coordinate by substituting \( x = 0 \) back into the function:
\[ f(0) = (0)^2 - 4 = -4 \]
Therefore, the y-coordinate of the vertex is \( -4 \). Combining both coordinates gives us the vertex \( (0, -4) \).
parabolic equations
Parabolic equations describe the shape and orientation of the parabolas. The graph of a quadratic function \( f(x) = ax^2 + bx + c \) is a parabola. The position and shape of this parabola are influenced by the coefficients \( a \), \( b \), and \( c \):
  • If \( a > 0 \), the parabola opens upwards.
  • If \( a < 0 \), the parabola opens downwards.
  • \( b \) affects the horizontal placement of the vertex.
  • \( c \) affects the vertical placement of the parabola.

The vertex represents the highest or lowest point on the graph of the parabola, depending on its direction:
  • For an upward-opening parabola, the vertex is the minimum point.
  • For a downward-opening parabola, the vertex is the maximum point.

In the original exercise, our parabolic equation \( f(x) = x^2 - 4 \) yields an upward-opening parabola because \( a > 0 \) (\( a = 1 \)). We identified the vertex at \( (0, -4) \), meaning this vertex is the lowest point on our parabola.

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