Problem 51 - Algebra 1

Question

Sketch the graph of the function. $$y=x^{2}$$

Step-by-Step Solution

Answer

The graph of the function \(y = x^{2}\) is a parabola with its vertex at the origin, (0,0), and opens upward.

Step 1: Identifying the type of function

The function \(y = x^{2}\) is a basic example of a quadratic function. The general form of a quadratic function is \(y = ax^{2} + bx + c\), but in this case, \(a = 1\), \(b = 0\) and \(c = 0\). The function makes a 'U'-shaped curve called a parabola.

Step 2: Plot suitable values

Choose a range of x-values (including negative, positive, and zero), and find the corresponding y-values by substituting each x-value into the function. In this case, the chosen x-values could be \(-2,-1,0,1,2\). The corresponding y-values for these x-values would respectively be \(4,1,0,1,4\).

Step 3: Sketch the graph

Next plot these pairs of values (x,y) on the graph as coordinate points. Joining these points will give a smooth curve of the function \(y = x^{2}\). The parabola has its vertex (its lowest point, because it opens upward) at (0,0).

Key Concepts

Parabola

A parabola is a symmetrical, U-shaped curve that represents the graph of a quadratic function. It's named after the geometric shape that forms when a plane slices through a cone parallel to its side. Quadratic functions like \(y = x^2\) have a standard form equation of \(y = ax^2 + bx + c\). The parabola will open upwards if \(a\) is positive and downwards if \(a\) is negative. Parabolas have unique characteristics such as:

• A single axis of symmetry, which in this case is the y-axis.
• The curve is symmetric with respect to this axis.
• The lowest or highest point of a parabola is known as the vertex.

Understanding the shape and direction of a parabola is essential in graphing quadratic functions effectively.

Graph Sketching

Graph sketching involves visually representing mathematical functions, such as quadratic functions, on a coordinate plane. It's a powerful tool that conveys complex algebraic relationships in an easy-to-understand way. To sketch the graph of \(y = x^2\), follow these steps:

• Identify the key features, like the axis of symmetry and the vertex.
• Choose relevant x-values, such as \(-2, -1, 0, 1,\) and \(2\), and compute their corresponding y-values: 4, 1, 0, 1, and 4.
• Plot these (x, y) coordinates on the graph.
• Draw a smooth curve through the points to complete the parabola.

By systematically performing these steps, it becomes easier to conceptualize how the quadratic function behaves across different x-values.

Function Plot

In mathematics, a function plot is a graphical representation of a function's behavior. For a quadratic function like \(y = x^2\), the plot will showcase the shape of the parabola, helping us visualize how the function behaves. Key aspects of plotting the quadratic function \(y = x^2\):

• The graph is centered around the origin (0,0).
• All points on the graph are equidistant from the y-axis due to symmetry.
• As \(x\) increases or decreases, \(y\) values rise, shown by the parabola opening upwards.

Function plots reveal the relationship between variables, allowing us to understand changes and predict values within a specific range.

Vertex of a Parabola

The vertex of a parabola is its turning point, a critical feature where the curve changes direction. In the quadratic function \(y = x^2\), the vertex is at the point (0,0), also known as the origin. Understanding the vertex is important because:

• It is the minimum or maximum point of the parabola.
• The vertex provides a reference point for the graph's symmetry.
• It's the lowest point on the graph for upward-opening parabolas like \(y = x^2\).

Calculating the vertex of a general quadratic \(y = ax^2 + bx + c\) involves finding \(x = -\frac{b}{2a}\). For \(y = x^2\), this is 0, placing the vertex right on the origin.