Problem 24 Graph each quadratic function, a... [FREE SOLUTION]

Chapter 8: Problem 24

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

Short Answer

Expert verified

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

Step by step solution

- 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.

- 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: \[ (0, -1) \].

- 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: \[ x = 0 \].

- 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: \[ (0, -1) \].

- 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.

- 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.

- State the Domain

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

- 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: \[ (-\infty, -1] \].

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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 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.

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