Problem 24 - Algebra for College Students

Question

Graph each quadratic function, and state its domain and range. $$g(x)=-x^{2}-1$$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\). Range: \((-\infty, -1]\). Vertex: (0, -1).

Step 1 - Identify the Quadratic Function

The quadratic function given is \[ g(x) = -x^2 - 1 \]. This is a downward-opening parabola because the coefficient of \(x^2\) is negative.

Step 2 - Find the Vertex

The vertex form of a quadratic function is \[ g(x) = a(x-h)^2 + k \]. In this case: \[ g(x) = -x^2 - 1 \]. The vertex \( (h, k) \) is at: \[ (0, -1) \].

Step 3 - Identify the Axis of Symmetry

The axis of symmetry is a vertical line that goes through the vertex. For this function, the axis of symmetry is: \[ x = 0 \].

Step 4 - Determine the Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. To find it, set \( x = 0 \): \[ g(0) = -0^2 - 1 = -1 \]. So, the y-intercept is: \[ (0, -1) \].

Step 5 - Determine the X-Intercepts

To find the x-intercepts, set \( g(x) = 0 \): \[ -x^2 - 1 = 0 \] This simplifies to: \[ -x^2 = 1 \] \[ x^2= -1 \]. Since there are no real solutions for \( x \) (the square root of a negative number is not real), there are no x-intercepts.

Step 6 - Sketch the Graph

Using the vertex \( (0, -1) \), the axis of symmetry \( x = 0 \), and the y-intercept \( (0, -1) \), sketch the parabola opening downward.

Step 7 - State the Domain

The domain of any quadratic function is all real numbers. So, the domain is: \[ (-\infty, \infty) \].

Step 8 - State the Range

The range is the set of all possible values of \( g(x) \). Since the parabola opens downward with a maximum value at the vertex \( y = -1 \), the range is: \[ (-\infty, -1] \].

Key Concepts

quadratic function

A quadratic function is a type of polynomial function often written in the form \(f(x) = ax^2 + bx + c\). In our example, the quadratic function is \(g(x) = -x^2 - 1\).

vertex

The vertex is one of the key points of a quadratic function. It forms the peak or the lowest point of the parabola, known as either the maximum or minimum point.

axis of symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror images. This line always passes through the vertex.

y-intercept

The y-intercept is where the graph crosses the y-axis. This occurs when \(x\) equals 0.

x-intercepts

The x-intercepts are the points where the graph crosses the x-axis. This happens when \(g(x)\) equals 0.

domain and range

The domain of a quadratic function includes all real numbers, from negative infinity to positive infinity. The range of a parabola opening downward is from negative infinity up to the y-value of the vertex, and the range of a parabola opening upward is from the y-value of the vertex to positive infinity.