Problem 22 - Algebra 2 and Trigonometry

Question

a. Sketch the graph of \(y=x^{2}\) . b. Sketch the graph of \(y=(x+2)^{2}\) c. Sketch the graph of \(y=(x-3)^{2}\) d. Describe the graph of \(y=(x+a)^{2}\) in terms of the graph of \(y=x^{2}\) e. What transformation maps \(y=x^{2}\) to \(y=(x+a)^{2} ?\)

Step-by-Step Solution

Answer

The transformations are horizontal shifts: left for \( (x+2)^2 \), right for \( (x-3)^2 \). Graph \( y = (x+a)^2 \) is a horizontal shift of \( y = x^2 \) by \(-a\) units.

Step 1: Sketch the graph of y = x^2

The graph of the function \( y = x^2 \) is a parabola that opens upwards. The vertex of this parabola is at the origin, \( (0,0) \), and it is symmetric about the y-axis. The graph passes through points such as \( (1,1) \), \( (-1,1) \), \( (2,4) \), and \( (-2,4) \).

Step 2: Sketch the graph of y = (x+2)^2

The graph of \( y = (x+2)^2 \) is a horizontal shift of the graph \( y = x^2 \). It is shifted 2 units to the left. The vertex of this parabola is \( (-2,0) \), and the parabola remains symmetric about the line \( x = -2 \).

Step 3: Sketch the graph of y = (x-3)^2

The graph of \( y = (x-3)^2 \) is another horizontal shift of the graph \( y = x^2 \). It is shifted 3 units to the right. The vertex of this parabola is \( (3,0) \), and the parabola remains symmetric about the line \( x = 3 \).

Step 4: Describe the graph of y = (x+a)^2

For the graph \( y = (x+a)^2 \), the graph of \( y = x^2 \) is horizontally shifted by \(-a\) units. If \( a > 0 \), the graph shifts left; if \( a < 0 \), the graph shifts right. This means the vertex of the parabola is at \( (-a, 0) \).

Step 5: Describe the transformation from y = x^2 to y = (x+a)^2

The transformation to map \( y = x^2 \) to \( y = (x+a)^2 \) is a horizontal translation. Specifically, it is a shift of \(-a\) units. This means the graph moves left by \( a \) units if \( a > 0 \), or right by \( a \) units if \( a < 0 \).

Key Concepts

Parabola

A parabola is a U-shaped curve that can either open upwards or downwards. It is a graph of a quadratic function, which means it represents functions of the form \( y = ax^2 + bx + c \).
A simple example of a parabola is the graph of \( y = x^2 \). This particular parabola is special because it opens upwards and has its vertex at the origin \( (0,0) \).
Being symmetric about the y-axis is another unique feature of this parabola; if you were to fold the graph along the y-axis, one side would perfectly align with the other.

• The basic formula of a parabola is \( y = x^2 \).
• It can either open upwards or downwards depending on the sign of the coefficient \( a \).
• The highest or lowest point, called the vertex, depends on the direction in which the parabola opens.

Vertex Form

The vertex form of a quadratic function allows us to directly identify and discuss its vertex, which is the peak or bottom of the parabola. It is written as \( y = a(x-h)^2 + k \), where \( (h,k) \) is the vertex.
Understanding the vertex form gives a quick way to determine how the parabola shifts and its vertex location.
When you write a quadratic function in vertex form, it makes it easier to see at a glance:

• The "h" shows the horizontal shift.
• The "k" shows the vertical shift.

For example, in \( y = (x+2)^2 \), you can spot straight away that the vertex is at \( (-2,0) \) because it tells us the graph is moved 2 units left.

Horizontal Shift

Horizontal shifts deal with moving the parabola side to side while keeping its shape the same.
This shift is accomplished by changing the \( x \) variable in the function. If you add or subtract a number to \( x \), the entire graph will move accordingly.

• Adding a positive number inside the function results in the graph moving left.
• Subtracting a number results in the graph moving right.

For instance, comparing \( y = x^2 \) and \( y = (x+3)^2 \), the latter is moved 3 units to the left. Notably, the shape or direction of the parabola remains the same; only its position changes.

Symmetry

Symmetry in parabolas indicates that one side of the parabola mirrors the other. Parabolic symmetry is typically about a vertical line going through its vertex, called the axis of symmetry.
The standard form, \( y = x^2 \), is symmetric around the y-axis.
This can be understood easily by visualizing how, for every point \( (x, y) \) on the parabola, there’s a corresponding \( (-x, y) \) point on the opposite side of the axis.

• The axis of symmetry of a parabola \( y = (x-h)^2 + k \) is the line \( x = h \).
• It ensures the graph is balanced around this axis, maintaining equity on both sides.

Understanding symmetry helps in sketching the parabola and predicting its shape from different equations.