Question

The equation \(y=(x-c)^{2}\) defines a family of parabolas, one parabola for each value of \(c .\) On one set of coordinate axes, graph the members of the family for \(c=0, c=3\), and \(c=-2\)

Short answer

Graph parabolas \(y = x^2\), \(y = (x - 3)^2\), and \(y = (x + 2)^2\) with vertices at \((0,0)\), \((3,0)\), and \((-2,0)\) respectively.

Step-by-step solution

Understand the General Form

The given equation is a parabola in the form \(y = (x - c)^2\). This is a standard parabola that opens upwards, with its vertex at the point \((c, 0)\).

Identify Values of c

The problem gives three specific values for \(c\): \(c = 0\), \(c = 3\), and \(c = -2\). Each value of \(c\) will shift the vertex of the parabola horizontally along the x-axis.

Equation for c = 0

For \(c = 0\), the equation becomes \(y = (x - 0)^2\) or simply \(y = x^2\). This parabola has its vertex at \((0, 0)\).

Equation for c = 3

For \(c = 3\), the equation becomes \(y = (x - 3)^2\). This parabola has its vertex at \((3, 0)\).

Equation for c = -2

For \(c = -2\), the equation becomes \(y = (x + 2)^2\). This parabola has its vertex at \((-2, 0)\).

Graph the Parabolas

On the same set of coordinate axes, plot each of the parabolas. For \(y = x^2\), the curve is symmetric about the y-axis with its vertex at \((0, 0)\). For \(y = (x - 3)^2\), the curve is shifted 3 units to the right, with its vertex at \((3, 0)\). For \(y = (x + 2)^2\), the curve is shifted 2 units to the left, with its vertex at \((-2, 0)\).

Key concepts

Parabola

A parabola is a U-shaped curve that is symmetric along its axis. The general equation of a parabola is given by either \(y = ax^2 + bx + c\) or \(y = (x - h)^2 + k\). In this exercise, we specifically address the equation \(y = (x - c)^2\), which represents a parabola that opens upwards. The most distinctive feature of a parabola is its shape, and it can open either upwards or downwards based on the sign of the coefficient of the squared term.
The vertex and direction of the parabola are essential aspects. For the equation \(y = (x - c)^2\), the parabola always opens upwards since the squared term is positive. Understanding these properties helps in graphing and analyzing the parabola's behavior.

Vertex

The vertex is a crucial point on the parabola representing its maximum or minimum value. For the equation \(y = (x - c)^2\), the vertex is at the point \((c, 0)\). This is because the parabola is translated horizontally by the value of \(c\).
For example:

• When \(c = 0\), the vertex is at (0, 0).
• When \(c = 3\), the vertex is at (3, 0).
• When \(c = -2\), the vertex is at (-2, 0).

By changing the value of \(c\), we shift the parabola along the x-axis. The vertex also signifies the point where the parabola changes direction, making it important for understanding its geometrical properties.

Coordinate Axes

The coordinate axes consist of the x-axis (horizontal) and the y-axis (vertical). These axes help us graph functions, including parabolas, by providing a reference framework.
When graphing parabolas like \(y = (x - c)^2\), we plot points on the coordinate plane:

• The x-axis helps us understand horizontal shifts, as indicated by the changes in the value of \(c\).
• The y-axis helps us see how the function values (outputs) vary with different x-values.

For instance, in this exercise, we graph parabolas for different values of \(c\) using the same set of coordinate axes. This helps in visualizing how each parabola shifts horizontally based on its vertex.

Graphing Parabolas

Graphing parabolas involves plotting points and sketching the curve based on its equation. For the equation \(y = (x - c)^2\), each parabola can be graphed by following these steps:

• Identify the vertex \((c, 0)\).
• Determine symmetrical points about the vertex.
• Plot a few key points on either side of the vertex.
• Draw the U-shaped curve passing through these points, ensuring symmetry about the vertex.

In this exercise, for the values \(c = 0, c = 3,\) and \(c = -2\), the parabolas can be graphed as follows:

• For \(y = x^2\), the vertex is at (0, 0), and the graph is symmetric about the y-axis.
• For \(y = (x - 3)^2\), the vertex is at (3, 0), and the graph is shifted right.
• For \(y = (x + 2)^2\), the vertex is at (-2, 0), and the graph is shifted left.

By graphing these, we see how changing \(c\) alters the position of the parabola on the coordinate plane.