Question

Find (a) The domain. (b) The range. $$ y=x^{2} $$

Short answer

Question: Determine the domain and range of the quadratic function \(y = x^2\). Answer: (a) The domain of the function is \((-\infty, \infty)\). (b) The range of the function is \([0, \infty)\).

Step-by-step solution

Analyze the function

We have the quadratic function \(y = x^2\). This function is a parabola that opens upwards, with its vertex at the origin (0,0). Since it is a quadratic function, it is defined for all real numbers as there is no restriction on the input values.

Find the domain

Since there is no restriction on the input values, and a quadratic function is defined for all real numbers, the domain of the function is all real numbers. We can write this in interval notation as \((-\infty, \infty)\).

Find the range

To find the range, we need to look at the output values of the function. Since the parabola opens upwards, the minimum value of y will be achieved at the vertex. In this case, the vertex is at the origin (0,0) and as x increases or decreases, the value of y will always be greater than or equal to zero. Thus, the range of the function will consist of all non-negative real numbers. We can write this in interval notation as \([0, \infty)\). Final answer: (a) The domain of the function is \((-\infty, \infty)\). (b) The range of the function is \([0, \infty)\).

Key concepts

Domain and Range

When dealing with quadratic functions like \(y = x^2\), understanding the domain and range is crucial. The domain refers to all possible input values (x-values) for which the function is defined. In the case of \(y = x^2\), there are no restrictions on the input values because you can plug any real number into the function. Therefore, the domain is all real numbers, expressed in interval notation as \((-ablafty, \infty)\).

The range pertains to all possible output values (y-values) that the function can produce. For \(y = x^2\), the lowest point on the graph is at the vertex, which is (0,0). Since the parabola opens upwards, the y-values are never negative. They start at 0 and increase as x moves away from zero. Hence, the range is all non-negative real numbers, written in interval notation as \([0, \infty)\).

These concepts help us understand the behavior of the quadratic function, ensuring we're aware of what values we can expect \(y\) to take given any \(x\).

Parabolas

A parabola is a symmetrical curve resulting from a quadratic function like \(y = x^2\). Parabolas can open upwards or downwards. In this function, the parabola opens upwards, meaning as you move away from the vertex, the y-values increase. This orientation is indicated by the positive coefficient of \(x^2\). Conversely, a negative coefficient would flip the parabola so it opens downwards.

Key Characteristics of Parabolas:

• Vertex: The point where the curve changes direction. For \(y = x^2\), the vertex is at (0,0).
• Axis of Symmetry: A vertical line that runs through the vertex, giving the parabola a mirror image on either side. Here, the axis of symmetry is \(x = 0\).
• Direction: Determined by the coefficient of the \(x^2\) term. Positive for upwards, negative for downwards.

Understanding these properties can help you sketch the graph and solve quadratic equations effectively. Knowing the direction and position of the parabola allows us to predict the range of values that \(y\) can take.

Vertex of a Parabola

The vertex of a parabola is a crucial point and serves as a turning point for the curve. In standard form, a quadratic function is written as \(y = ax^2 + bx + c\). For \(y = x^2\), the vertex is surprisingly simple: it's at the origin (0,0).

Importance of the Vertex:

• It's the minimum or maximum point on the parabola, depending on the direction the parabola opens.
• In the case of \(y = x^2\), since it opens upwards, the vertex is the minimum point.
• Understanding the vertex allows for easier calculation of the parabola's range and symmetry.

To find the vertex in more complex quadratic functions \(y = ax^2 + bx + c\), one can use the formula \(x = -\frac{b}{2a}\) to find the x-coordinate of the vertex. The y-coordinate is then found by substituting back into the original equation. Thus, the vertex is pivotal in understanding the behavior of quadratic functions, defining both the graph's principal point and the start of the range of possible y-values.