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A parabola is a U-shaped curve that can open either upwards or downwards. For the quadratic function \( f(x) = x^2 - 1 \), the parabola opens upwards because the coefficient of the \( x^2 \) term is positive. The general structure of a quadratic function is \( ax^2 + bx + c \), where 'a' determines the direction of the parabola. Here, \( a = 1 \), \( b = 0 \), and \( c = -1 \).

• The vertex of the parabola is the point where it changes direction, acting as the minimum or maximum point.
• For \( f(x) = x^2 - 1 \), the vertex is at \( (0, -1) \).
• The parabola intersects the y-axis at the point \( (0, -1) \), as \( c = -1 \).

The graph of this function is symmetric about the y-axis, meaning the left and right sides are mirror images.

Understanding the domain and range of a function is crucial in graph interpretation. For the quadratic \( f(x) = x^2 - 1 \), the domain is all real numbers, represented as \( (-\infty, \infty) \). This is because there are no restrictions on the values of \( x \) you can use in a quadratic.

• The domain indicates all the input values \( x \) for which the function is defined.
• For polynomials like \( f(x) = x^2 - 1 \), you can plug in any real number and get a valid output.

The range deals with all possible output values \( f(x) \). Since the vertex of the parabola is at \( (0, -1) \) and the parabola opens upwards, the minimum value the function can take is \( -1 \). This results in the range being \( [-1, \infty) \), meaning it includes all real numbers from \( -1 \) upwards.

Identifying intervals where a function is increasing or decreasing helps understand its behavior. To determine this for \( f(x) = x^2 - 1 \), we can use its derivative \( f'(x) = 2x \). The derivative provides the slope at any point \( x \):

• When \( f'(x) > 0 \), the function is increasing.
• When \( f'(x) < 0 \), the function is decreasing.

At the vertex \( (0, -1) \), the derivative is zero, indicating neither increase nor decrease.- For \( x < 0 \), \( f'(x) = 2x < 0 \), indicating that the function is decreasing.- For \( x > 0 \), \( f'(x) = 2x > 0 \), indicating that the function is increasing.Therefore, \( f(x) \) is decreasing in the interval \((-\infty, 0)\) and increasing in the interval \((0, \infty)\). This information reflects the overall shape and direction of the parabola around its vertex.