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Chapter 9: Problem 27

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. \(f(x)=x^{2}+4 x+3\)

Short Answer

Expert verified
Vertex: (0, -1). Axis of Symmetry: x = 0. Domain: (-鈭�, 鈭�). Range: [-1, 鈭�).

Step by step solution

01

Identify the General Form of the Parabola

The given function is in the form of a quadratic equation: \( f(x) = x^2 - 1 \). The general form of a quadratic function is \( f(x) = ax^2 + bx + c \). Here, \( a = 1 \), \( b = 0 \), and \( c = -1 \).
02

Determine the Vertex

The vertex of a parabola given by \( f(x) = ax^2 + bx + c \) is found using the formula: \( x = -\frac{b}{2a} \). For \( f(x) = x^2 - 1 \), \( b = 0 \) and \( a = 1 \). Thus, \( x = -\frac{0}{2(1)} = 0 \). Substituting \( x = 0 \) back into the function: \( f(0) = 0^2 - 1 = -1 \). Therefore, the vertex is at \( (0, -1) \).
03

Find the Axis of Symmetry

The axis of symmetry for a parabola \( f(x) = ax^2 + bx + c \) is given by the vertical line \( x = -\frac{b}{2a} \). Since \( x = 0 \) for this parabola, the axis of symmetry is \( x = 0 \).
04

Determine the Domain

The domain of a quadratic function \( f(x) = ax^2 + bx + c \) is all real numbers. So, the domain of \( f(x) = x^2 - 1 \) is \( (-\infty, \infty) \).
05

Determine the Range

The range of \( f(x) = ax^2 + bx + c \) depends on the direction of the parabola (opening up or down). Since \( a = 1 \) and is positive, the parabola opens upward. The minimum value of \( f(x) \) is at the vertex \( (0, -1) \), so the range is \( [-1, \infty) \).
06

Graph the Parabola

To graph the parabola, plot the vertex \( (0, -1) \) and sketch the curve opening upwards. Use additional points such as \((1, 0) \) and \((-1, 0)\) to help shape the parabola.

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Key Concepts

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

vertex
The vertex of a parabola is a fundamental point that describes where the curve changes direction. For the quadratic function \( f(x) = ax^2 + bx + c \), the vertex formula is \( x = -\frac{b}{2a} \). After finding this \( x \) value, substitute it back into the function to get the \( y \) value.
For instance, in the given function \( f(x) = x^2 - 1 \):
  • Identify \( a = 1 \) and \( b = 0 \).
  • Calculate \( x = -\frac{0}{2(1)} = 0 \).
  • Substitute \( x = 0 \) into the function: \( f(0) = 0^2 - 1 = -1 \).
The vertex is therefore at \( (0, -1) \), indicating the minimum point on the curve.
axis of symmetry
The axis of symmetry is a vertical line that splits the parabola into two identical halves. For any quadratic function \( f(x) = ax^2 + bx + c \), this line always passes through the vertex and its formula is \( x = -\frac{b}{2a} \).
Taking the same example, \( f(x) = x^2 - 1 \):
  • Identify the vertex at \( x = 0 \).
  • Thus, the axis of symmetry is the vertical line \( x = 0 \).
This line helps in understanding the symmetry of the parabola's graph.
domain
The domain of a function represents all the possible \( x \) values that can be input into the function. For quadratic functions of the form \( f(x) = ax^2 + bx + c \), the domain is always all real numbers because you can input any real number into the function.
For the quadratic function \( f(x) = x^2 - 1 \), the domain remains:
  • \( (-\infty, \infty) \).
This means the function can accept any real number as a valid input.
range
The range of a function encompasses all the possible \( y \) values that the function can output. For a quadratic function like \( f(x) = ax^2 + bx + c \), the range is determined by the direction in which the parabola opens (upward or downward).
In our example, \( f(x) = x^2 - 1 \):
  • The coefficient \( a = 1 \) is positive, so the parabola opens upward.
  • The lowest point is the vertex, which is \( (0, -1) \).
Hence, the range is:
  • \( [-1, \infty) \), meaning all \( y \) values starting from \( -1 \) and extending to infinity.
quadratic function
A quadratic function is a type of polynomial function with the highest degree term being squared. Its general form is \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.
  • The graph of a quadratic function is a parabola.
  • It can open either upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)).
  • Key features include the vertex, axis of symmetry, domain, and range.
Understanding each part of a quadratic function helps in graphing and analyzing it effectively, as seen in the given example \( f(x) = x^2 - 1 \), where we identified the vertex as \( (0, -1) \), the axis of symmetry as \( x = 0 \), a domain of all real numbers, and a range of \( [-1, \infty) \).

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