Question

What vertical translation is applied to \(y=x^{2}\) if the transformed graph passes through the point (4,19)\(?\)

Short answer

The vertical translation is 3 units up.

Step-by-step solution

Identify the Original Equation

The given function is the basic quadratic equation: y = x^2

Understand Vertical Translation

A vertical translation adds a constant value, c, to the original function. This can be represented as: y = x^2 + c

Substitute the Point into the Translated Equation

Given the point (4, 19), substitute x = 4 and y = 19 into the translated equation: 19 = 4^2 + c

Solve for the Translation Constant

Solve the equation 19 = 16 + c for c: c = 19 - 16 c = 3

Conclude the Vertical Translation

The vertical translation applied to the original function y = x^2 is an upward shift by 3 units.

Key concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation in which the highest power of the variable is 2. The general form of a quadratic equation is:
\[ y = ax^2 + bx + c \]
In this equation:

• a represents the coefficient of the quadratic term (x^2),
• b is the coefficient of the linear term (xyi)
• c is the constant term.

For the exercise given, the quadratic equation is in its simplest form, \( y = x^2 \), where \(a = 1, b = 0, c = 0\). This represents a parabola that opens upwards. Understanding the general form of a quadratic equation helps in comprehending transformations, including vertical translations.

Transformation of Functions

Transforming functions is a way to shift or stretch the graph of a function. There are multiple types of transformations, including:

• Vertical translations (moving up or down),
• Horizontal translations (moving left or right),
• Reflections (flipping over an axis),
• Scaling (stretching or compressing).

In the given exercise, we focus on vertical translation. For the function \( y = x^2 \), a vertical translation involves adding or subtracting a constant value, \( c \). This shifts the graph of \( y = x^2 \) upwards by \( c \) units if \( c \) is positive, and downwards by \( c \) units if \( c \) is negative. The translated function then becomes:
\[ y = x^2 + c \]
In this scenario, substituting \( x = 4 \) and \( y = 19 \) gives us the following equation: \( 19 = 4^2 + c \), which is then solved for \( c \)

Solving for Constants

In many problems, you may need to determine the value of a constant that satisfies certain conditions. In this exercise, we find the vertical translation constant \( c \). Follow these steps:

1. Start with the translated function \( y = x^2 + c \) and substitute the given point into the equation.
2. For the point (4, 19): \( 19 = 4^2 + c \) simplifies to \( 19 = 16 + c \).
3. Isolate \( c \) by subtracting 16 from both sides: \( c = 19 - 16 \).
4. Simplify the result to find \( c = 3 \).

This constant \( c \) indicates that the graph of the quadratic function \( y = x^2 \) is shifted vertically upward by 3 units. Solving for constants is a crucial skill as it allows you to customize functions based on specific conditions or points they must pass through.