Problem 45 - College Algebra

Question

In Exercises \(45-48,\) give the domain and the range of each quadratic function whose graph is described. The vertex is \((-1,-2)\) and the parabola opens up.

Step-by-Step Solution

Answer

The domain of the given quadratic function is \((-∞, ∞)\) and the range is \([-2, ∞)\).

Step 1: Identify the Domain of the Quadratic Function

The domain of a quadratic function is always all real numbers. This is due to the fact that the graph of the function can stretch infinitely in both positive and negative x-directions. Hence, domain for this function is all real numbers, represented as \((-∞, ∞)\).

Step 2: Determine the Direction of Opening of the Parabola

The exercise states that the parabola opens upwards. This means that the minimum value the function reaches is at the vertex and then it continues to increase from there.

Step 3: Identify the Range of the Quadratic Function

Since the parabola opens upwards, the range of the function will be the y-coordinate of the vertex and all values greater than it, as that is the lowest point on the graph. The vertex given in this problem is \((-1, -2)\), hence the range of this function would be \([-2, ∞)\).

Key Concepts

Domain and Range

The domain and range are crucial aspects of understanding quadratic functions. When we talk about the **domain**, we focus on all possible inputs for the function—essentially, the x-values.
A quadratic function, represented as a parabola, always has a domain of all real numbers. This is because a parabola extends infinitely in both the positive and negative x-directions. Therefore, in interval notation, the domain is written as

• offset{5ex}(-∞, ∞)

Now, the **range** of a quadratic function varies based on the direction the parabola opens (either upwards or downwards). If the parabola opens upwards, as in this exercise example, the range includes all the y-values starting from the y-coordinate of the vertex going up to infinity.
Because the parabola in this scenario opens upwards and the vertex is at (-1, -2), the lowest y-value is -2. Thus, the range is expressed as

• otset{5ex}[-2, ∞)

Parabolas

Parabolas are the U-shaped graphs of quadratic functions, and they have several notable characteristics. One defining trait is the direction in which the parabola opens, either upwards or downwards. This direction is determined by the leading coefficient (the coefficient before the squared term) in the quadratic equation.
- If this coefficient is positive, the parabola opens upwards, resembling a smile. - If negative, it opens downwards, resembling a frown.
Parabolas also have a symmetrical property. They are symmetric along a vertical line known as the axis of symmetry. This line passes through the **vertex** of the parabola, neatly dividing it into two mirror-image halves. This symmetry is another helpful feature for visualizing and understanding parabolas more deeply.

Vertex of a Parabola

The vertex is a significant point on the parabola, serving as the turning point of the curve. Whether a parabola opens upwards or downwards, the vertex is always at its peak or trough. Its coordinates can often be found in standard quadratic function forms or calculated using the formula for the vertex:

• For the equation ( ax^2 + bx + c ) , the x-coordinate of the vertex is offset{5ex} (-b/(2a)

Once you determine the x-coordinate, substitute it back into the equation to find the y-coordinate. For a quadratic function opening upwards, as seen in the exercise:
- The vertex (-1, -2) indicates the parabola reaches its minimum point at y = -2. The vertex not only helps in sketching the parabola but also plays a key role in determining the range of the function.