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Introduction

In mathematics, quadratic functions are a fundamental concept in algebra and geometry. They are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, and the motion of an object under the influence of a force. One of the most important forms of a quadratic function is the vertex form, which provides valuable information about the graph of the function. In this article, we will learn how to convert a quadratic function to vertex form and identify the transformations of its graph.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by:

$$h(x) = (x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola. The vertex form is useful because it allows us to easily identify the transformations of the graph of the function.

Converting a Quadratic Function to Vertex Form

To convert a quadratic function to vertex form, we need to complete the square. Let's consider the quadratic function:

$$h(x) = x^2 - 4x - 3$$

Our goal is to rewrite this function in vertex form. To do this, we need to complete the square.

Step 1: Factor out the coefficient of the x^2 term

The coefficient of the $x^2$ term is 1, so we can factor it out:

$$h(x) = (x^2 - 4x) - 3$$

Step 2: Add and subtract the square of half the coefficient of the x term

The coefficient of the $x$ term is -4, so half of it is -2. The square of -2 is 4, so we add and subtract 4:

$$h(x) = (x^2 - 4x + 4) - 3 - 4$$

Step 3: Simplify the expression

Now we can simplify the expression:

$$h(x) = (x - 2)^2 - 7$$

And that's it! We have successfully converted the quadratic function to vertex form.

Identifying Transformations

Now that we have the quadratic function in vertex form, we can easily identify the transformations of its graph. The vertex form of a quadratic function is given by:

$$h(x) = (x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola. The vertex form tells us that the graph of the function is a parabola that opens upward or downward, with its vertex at the point $(h, k)$.

Horizontal Shift

The value of $h$ in the vertex form tells us about the horizontal shift of the graph. If $h$ is positive, the graph is shifted to the right by $h$ units. If $h$ is negative, the graph is shifted to the left by $|h|$ units.

Vertical Shift

The value of $k$ in the vertex form tells us about the vertical shift of the graph. If $k$ is positive, the graph is shifted upward by $k$ units. If $k$ is negative, the graph is shifted downward by $|k|$ units.

Reflection

If the coefficient of the $x^2$ term is negative, the graph is reflected across the x-axis.

Stretching and Compressing

If the coefficient of the $x^2$ term is not 1, the graph is stretched or compressed vertically.

Example

Let's consider the quadratic function:

$$h(x) = (x - 3)^2 + 2$$

To identify the transformations of its graph, we need to look at the vertex form. The value of $h$ is 3, which means that the graph is shifted to the right by 3 units. The value of $k$ is 2, which means that the graph is shifted upward by 2 units.

Conclusion

In this article, we learned how to convert a quadratic function to vertex form and identify the transformations of its graph. The vertex form of a quadratic function is given by:

$$h(x) = (x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola. We also learned how to identify the horizontal shift, vertical shift, reflection, and stretching/compressing of the graph of a quadratic function.

Practice Problems

1. Convert the quadratic function $h(x) = x^2 + 6x + 8$ to vertex form.
2. Identify the transformations of the graph of the quadratic function $h(x) = (x - 2)^2 - 3$.
3. Convert the quadratic function $h(x) = x^2 - 2x - 6$ to vertex form.
4. Identify the transformations of the graph of the quadratic function $h(x) = (x + 3)^2 + 1$.

Answer Key

1. $h(x) = (x + 3)^2 + 2$
2. The graph is shifted to the right by 2 units and downward by 3 units.
3. $h(x) = (x - 1)^2 - 7$
4. The graph is shifted to the left by 3 units and upward by 1 unit.
   Quadratic Functions in Vertex Form: Q&A =============================================

Introduction

In our previous article, we learned how to convert a quadratic function to vertex form and identify the transformations of its graph. In this article, we will answer some frequently asked questions about quadratic functions in vertex form.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by:

$$h(x) = (x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola.

Q: How do I convert a quadratic function to vertex form?

A: To convert a quadratic function to vertex form, you need to complete the square. Here are the steps:

1. Factor out the coefficient of the $x^2$ term.
2. Add and subtract the square of half the coefficient of the $x$ term.
3. Simplify the expression.

Q: What is the significance of the value of h in the vertex form?

A: The value of $h$ in the vertex form tells us about the horizontal shift of the graph. If $h$ is positive, the graph is shifted to the right by $h$ units. If $h$ is negative, the graph is shifted to the left by $|h|$ units.

Q: What is the significance of the value of k in the vertex form?

A: The value of $k$ in the vertex form tells us about the vertical shift of the graph. If $k$ is positive, the graph is shifted upward by $k$ units. If $k$ is negative, the graph is shifted downward by $|k|$ units.

Q: How do I identify the transformations of the graph of a quadratic function?

A: To identify the transformations of the graph of a quadratic function, you need to look at the vertex form. The value of $h$ tells you about the horizontal shift, and the value of $k$ tells you about the vertical shift.

Q: What is the difference between a quadratic function in standard form and vertex form?

A: A quadratic function in standard form is given by:

$$h(x) = ax^2 + bx + c$$

where $a$, $b$, and $c$ are constants. A quadratic function in vertex form is given by:

$$h(x) = (x - h)^2 + k$$

where $(h, k)$ is the vertex of the parabola.

Q: Can I use the vertex form to graph a quadratic function?

A: Yes, you can use the vertex form to graph a quadratic function. The vertex form tells you about the vertex of the parabola, which is the highest or lowest point on the graph.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you need to look at the vertex form. The vertex is given by the point $(h, k)$.

Q: Can I use the vertex form to find the x-intercepts of a quadratic function?

A: Yes, you can use the vertex form to find the x-intercepts of a quadratic function. The x-intercepts are given by the points where the graph intersects the x-axis.

Q: How do I use the vertex form to find the y-intercept of a quadratic function?

A: To find the y-intercept of a quadratic function, you need to substitute $x = 0$ into the vertex form and solve for $y$.

Conclusion

In this article, we answered some frequently asked questions about quadratic functions in vertex form. We hope that this article has been helpful in understanding the concept of quadratic functions in vertex form and how to use it to graph and analyze quadratic functions.

Practice Problems

1. Convert the quadratic function $h(x) = x^2 + 6x + 8$ to vertex form.
2. Identify the transformations of the graph of the quadratic function $h(x) = (x - 2)^2 - 3$.
3. Find the vertex of the quadratic function $h(x) = (x + 3)^2 + 2$.
4. Find the x-intercepts of the quadratic function $h(x) = (x - 1)^2 - 7$.
5. Find the y-intercept of the quadratic function $h(x) = (x + 2)^2 + 1$.

Answer Key

1. $h(x) = (x + 3)^2 + 2$
2. The graph is shifted to the right by 2 units and downward by 3 units.
3. The vertex is $(h, k) = (-3, 2)$.
4. The x-intercepts are $(1, 0)$ and $(-1, 0)$.
5. The y-intercept is $(0, 1)$.