A quadratic function is an equation where a variable is raised to the second power. Quadratic functions can be manipulated and translated in a variety of ways.

In this section, we will look at the translation from the graph y = 6×2 to the graph of y = 6(x + 1)2 and discuss which phrase best describes this transformation.

### Understanding the Basics of Quadratic Functions

A quadratic function is a mathematical function that can be described by an equation of y = axÂ² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola.

In the equation y = 6(x + 1)Â², the phrase “the graph of y = 6(x + 1)Â² is y = 6xÂ² translated one unit to the left” best describes the translation from the graph y = 6xÂ² to y = 6(x + 1)Â².

A translation is a type of transformation that shifts the graph of a function horizontally, vertically, or both, without changing its shape. In this case, the graph has been shifted to the left by one unit, represented by the expression (x + 1).

Understanding the basics of quadratic functions and translations can help you analyze and graph different types of functions and solve real-world problems involving quadratic equations.

Pro Tip: When graphing a quadratic function, always find the vertex, the axis of symmetry, and the x- and y-intercepts to get a better idea of the shape and behavior of the function.

### Recognizing the Different Forms of Quadratic Equations

Quadratic equations can be presented in different forms, including standard, vertex, and factored forms. Each form has unique advantages, depending on the context, and different forms can be easily converted to one another.

Form Equation

Vertex form: y = a(x – h)^2 + k, where the vertex is at (h,k)

In the example of y = 6(x + 1)^2, it is clear that the vertex is at (-1,0), indicating a translation of one unit to the left (as the value of h is negative). Additionally, the value of “a” is positive, indicating that the parabola opens upwards.

Understanding the different forms of quadratic equations is essential for solving problems in algebra, calculus, and physics. Moreover, recognizing quadratic functions’ translation (or other transformations) is fundamental in sketching their graphs and analyzing their features.

Pro Tip: To convert a quadratic equation from standard form to vertex form, complete the square by halving the coefficient of x and squaring the result.

### Graphing Quadratic Functions

When graphing quadratic functions, it is important to identify the key features of the function, such as the vertex, axis of symmetry, and direction of opening.

Describing a quadratic function requires identifying how the graph has been transformed from the parent function. For example, consider the quadratic function y = 6x^2 and the transformed function y = 6(x+1)^2. The phrase “left one unit” best describes the translation from the graph y = 6x^2 to the graph of y = 6(x+1)^2. This is because the transformation (x+1) shifts the graph horizontally to the left by one unit.

Understanding how to describe a quadratic function helps easily interpret the graph and its features. It is also essential in solving real-world problems that involve quadratic functions, such as finding the maximum height of a ball thrown in the air.

## Translating a Quadratic Function

The phrase “vertical shift” is commonly used to describe the change from one graph to another. In this context, the phrase references a vertical translation of the graph. This means that the graph of the function y = 6×2 is transformed into the graph y = 6(x + 1)2. Let’s take a closer look at the difference between these two graphs.

### Identifying the Vertex and Axis of Symmetry

The vertex of a quadratic function is the parabola’s highest or lowest point, depending on whether the function opens upward or downward. The axis of symmetry is the vertical line that passes through the vertex of the parabola and divides it into two halves.

To identify the vertex of a quadratic function in standard form, y = ax2 + bx + c, use the formula x = -b/2a to find the x-coordinate of the vertex, and substitute it into the function to find the y-coordinate.

For the given example of a quadratic function, y = 6(x + 1)2, the phrase “to the left by one unit” best describes the translation from the graph of y = 6×2 to the graph of y = 6(x + 1)2. The transformation involves shifting the entire parabola one unit to the left, without changing its shape or orientation.

Pro Tip: Mastering the technique of identifying the vertex and axis of symmetry of a quadratic function is crucial in understanding its properties such as maximum or minimum values, range, domain, and intercepts.

### Understanding Horizontal and Vertical Translations

In the context of graphing quadratic functions, understanding vertical and horizontal translations is crucial. A vertical translation involves moving the entire function graph up or down, while a horizontal translation involves moving the graph left or right.

In the example given, the translation from the graph of y = 6xÂ² to the graph of y = 6(x+1)Â² involves a horizontal translation of 1 unit to the left. This is because the “x+1” term in the new equation means that for any value of x, the corresponding point on the graph will be 1 unit to the left of the corresponding point on the original graph.

Understanding these translations is essential in graphing quadratic functions and evaluating how changes to the equation impact the shape and placement of the graph.

Pro tip: To remember which translation is which, think of the “v” in “vertical” going up and down, and the “h” in “horizontal” going left and right.

### Applying Transformations to Quadratic Equations

Transforming quadratic equations is an essential tool for analyzing and manipulating quadratic functions. In this context, applying transformations refers to modifying the variables in the equation to shift, stretch, or reflect the graph of a quadratic function.

The transformation from y = 6×2 to y = 6(x + 1)2 involves a horizontal translation of one unit to the left, indicated by the (x + 1). This transformation shifts the entire parabola of the quadratic function to the left, while keeping its shape and orientation the same.

Generally, transformations allow us to change essential characteristics of the parabola of the quadratic function to study its behavior and find its roots, maxima, minima, and intercepts. They also help us visualize how changing the variables in the equation affect the graph, which has real-life applications in modeling various phenomena.

## Which Phrase Best Describes the Translation From the Graph y = 6×2 to the Graph of y = 6(x + 1)2?

When looking at the given translations for the equation y = 6×2 to the graph of y = 6(x + 1)2, the phrase that best describes the translation is a “horizontal shift”. This is because the graph of y = 6×2 has been shifted 1 unit to the right. This can be seen when looking at the coordinates on the graph and the equation, and the changes in them. Let’s take a closer look at how this translation works.

### Breaking Down the Translation in y = 6x^2 to y = 6(x+1)^2

The translation from the graph y = 6x^2 to the graph of y = 6(x+1)^2 can be best described as a horizontal shift to the left by one unit. This is because adding the constant term “+1” within the parentheses of the quadratic expression shifts the entire parabola one unit towards the left of the y-axis. To see this, we can compare the vertex of both parabolas: the vertex of the original y = 6x^2 is (0,0) while the vertex of the translated parabola y = 6(x+1)^2 is (-1,0), which confirms the leftward shift.

It’s worth mentioning that the magnitude of the parabola’s stretch or compression, in this case multiplying the whole expression by 6, remains the same before and after translation, as it only affects the vertical scaling of the graph.

Pro tip: Understanding the effect of the translation parameter on the graph of quadratic functions can help you quickly and accurately sketch their graphs.

### Examining the Changes in the Graph

The translation from the graph of y = 6xÂ² to y = 6(x + 1)Â² results in a horizontal shift of the parabola to the left by 1 unit, without changing its shape or size.

To analyze the given translation, we can look at the vertex of each parabola. In y = 6xÂ², the vertex is at (0, 0). In y = 6(x + 1)Â², the vertex is at (-1, 0). The vertex moved horizontally to the left by 1 unit, the same amount as the value inside the parentheses, (x + 1).

It is important to note that the coefficient of xÂ² remains the same, so the shape and size of the parabola remain unchanged. Understanding these changes in the graph helps us to visualize the effect of translations on functions and to solve related mathematical problems with ease.

### Determining Which Phrase Best Describes the Translation

The translation of the quadratic function y = 6×2 to the graph of y = 6(x + 1)2 can best describe a horizontal shift of one unit to the left and an upward shift of the vertex.

The transformation of the function through the (x + 1) term maps the function one unit to the left, while the 2 term squares this value, leading to an upward shift of the vertex. Therefore, the translation phrase “one unit left and up” aptly describes the transformation in question.

## Additional Applied Problems

When given a quadratic function, there are a few key points to consider. For example, the graph of y = 6×2 can be translated to a graph of y = 6(x + 1)2. What does this change mean in terms of the original graph?

In this section, we will explore the answer to this question, discussing the effects of the transformation on the graph and equation.

### Solving Quadratic Equations with Translations

When solving quadratic equations using translations, it’s important to understand how different equation transformations affect the variables’ values. For example, in the given problem, the translation from the graph y=6x^2 to y=6(x+1)^2 refers to a horizontal shift of the graph one unit to the left.

To solve this problem, we need to understand that the value inside the brackets affects the horizontal shift of the graph, while the value outside the brackets affects its vertical stretch or compression.

The phrase “one unit to the left” indicates that we must subtract 1 from the original x-coordinate. Therefore, the correct graph translation is y=6(x-1)^2.

Understanding translations is beneficial when working with quadratic functions as it aids in graphing and analyzing their behaviors.

### Real-World Applications of Quadratic Equations and Translations

The phrase that best describes the translation from the graph y = 6x^2 to the graph of y = 6(x + 1)^2 is a horizontal translation to the left by one unit. Translations are essential to quadratic functions and have real-world applications, such as in projectile motion, engineering, and optimization problems.

For example, quadratic functions can be used to model the trajectory of a ball thrown in the air. By identifying key variables such as the initial velocity, angle of projection, and air resistance, quadratic equations can estimate the ball’s maximum height, time in the air, and landing distance.

Another example is in engineering, where quadratic functions can be used to model the strength of materials and predict their breaking point under different loads and stresses.

Translations can also optimize business operations by modeling production costs, revenue, and profit as quadratic functions and identifying the optimal output level that maximizes profits.

Overall, understanding the real-world applications of quadratic equations and translations is crucial in many fields and can help solve complex problems and make informed decisions.

### Analyzing Different Translations and Their Effects on the Graph

When analyzing different translations of quadratic functions, it is important to understand the effect of different phrases on the graph. For example, in this problem, we are comparing the graph of y = 6x^2 to the graph of y = 6(x + 1)^2.

The phrase “x + 1” in the second equation represents a horizontal translation of one unit to the left of the standard parabola. This means that the entire graph of y = 6(x + 1)^2 has been shifted one unit to the left compared to the graph of y = 6x^2. Specifically, the turning point of the graph has shifted from the origin (0,0) to the point (-1,0).

It is important to understand how different translations affect the shape and position of a quadratic graph, as this knowledge can be applied to real-world situations such as analyzing motion or predicting trends in data.