Problem 39 - Algebra 2

Question

The graph of each equation is to be translated 3 units right and 5 units up. Write each new equation. \(y=4 x^{2}\)

Step-by-Step Solution

Answer

The translated equation is \(y = 4(x - 3)^{2} + 5\).

Step 1: Translate the graph to the right

In a graphical interpretation, moving the graph 3 units to the right means that the value of x in the equation will decrease by 3. To represent this mathematically, replace every instance of x in the equation with (x - 3). This changes the equation \(y = 4x^{2}\) to \(y = 4(x - 3)^{2}\).

Step 2: Translate the graph upwards

Moving the graph 5 units up means that the value of y in the equation will increase by 5. To illustrate this numerically, add 5 to the entire equation, changing it from \(y = 4(x - 3)^{2}\) to \(y = 4(x - 3)^{2} + 5\). This completes the translation of the graph.

Key Concepts

Translating Functions

Translating functions involves shifting the entire graph of a function either horizontally or vertically, without altering its shape or orientation.
To translate a function:

• Horizontally: Adjust the x-values in the equation. Moving the function to the right involves subtracting a value from x, while moving it to the left involves adding to x.
• Vertically: Change the y-values by adding or subtracting a constant. To move a function up, a positive constant is added, whereas shifting it down requires subtracting a constant.

In our given exercise, the function is translated 3 units right, which modifies our x-values as \(x - 3\). This means replacing x with \(x - 3\) in the function's equation. To translate 5 units up, simply add 5 to the entire function, resulting in an equation of the form \ y = f(x) + 5 \.

Quadratic Functions

A quadratic function is a type of polynomial function where the highest degree of the variable is 2. It follows the form \[ f(x) = ax^2 + bx + c \]where:

• a, b, and c are constants;
• a \( eq \) 0;
• x is the variable.

The graph of a quadratic function is a parabola.
Depending on the coefficient a, the parabola can open either upward (when a > 0) or downward (when a < 0). The backbone of our example is the function \(y = 4x^2\), a simple form of a quadratic equation with a vertex at the origin (0, 0) and opens upwards because the coefficient 4 is positive.
When we translate this function, the shape remains a parabola, but its position on the coordinate plane changes. Applying the translations as described, the function becomes \(y = 4(x - 3)^2 + 5\), shifting both right and up from its original placement.

Graphing Equations

Graphing equations is a visual way of representing a mathematical function or relationship.
For quadratic functions, graphing primarily involves sketching the shape of a parabola. Here’s how you can graph a quadratic function effectively:

• Identify the vertex, which is the turning point of the parabola. For functions like \(y = ax^2\), the vertex is at the origin (0,0). For functions like our translated example \(y = 4(x - 3)^2 + 5\), the vertex becomes (3,5).
• Determine if the parabola opens upward or downward based on the sign of coefficient a. A positive a means it opens upwards.
• Select a few key points around the vertex to plot for accuracy. Substitute x-values into the equation to find corresponding y-values.
• Draw the axis of symmetry, a vertical line that goes through the vertex. This helps ensure that your parabola is symmetric.
• Connect the plotted points smoothly to form the parabola shape.

Remember, each translation alters the coordinates and visual position of the graph but not its fundamental parabola shape.