Find The Vertex, The Equation Of The Axis Of Symmetry, And The \[$ Y \$\]-intercept Of The Graph Of The Function:$\[ Y = -2x^2 + 8x - 5 \\]

Feb 26, 2025 by ADMIN 140 views

**Finding the Vertex, Axis of Symmetry, and y-Intercept of a Quadratic Function**

Introduction to Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $a$ is not equal to zero. These functions have a parabolic shape and can be used to model a wide range of real-world phenomena, such as the trajectory of a projectile or the spread of a disease.

The Vertex of a Quadratic Function

The vertex of a quadratic function is the highest or lowest point on the graph of the function. It is the point where the function changes from decreasing to increasing or vice versa. The vertex can be found using the formula $x = -\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic function.

Finding the x-Coordinate of the Vertex

To find the x-coordinate of the vertex, we need to plug in the values of $a$ and $b$ into the formula $x = -\frac{b}{2a}$ . In the given function $y = -2x^2 + 8x - 5$ , the values of $a$ and $b$ are $-2$ and $8$ , respectively.

a = -2
b = 8
x_vertex = -b / (2 * a)
print(x_vertex)

Running this code will give us the x-coordinate of the vertex, which is $x = 2$ .

Finding the y-Coordinate of the Vertex

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging this value into the original function. Substituting $x = 2$ into the function $y = -2x^2 + 8x - 5$ , we get:

y_vertex = -2 * (2)**2 + 8 * (2) - 5
print(y_vertex)

Running this code will give us the y-coordinate of the vertex, which is $y = 3$ .

The Equation of the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the quadratic function. It is the line that divides the graph of the function into two symmetrical parts. The equation of the axis of symmetry is given by $x = -\frac{b}{2a}$ , which is the same formula we used to find the x-coordinate of the vertex.

The y-Intercept of a Quadratic Function

The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. It is the value of $y$ when $x = 0$ . To find the y-intercept, we need to plug in $x = 0$ into the original function.

y_intercept = -2 * (0)**2 + 8 * (0) - 5
print(y_intercept)

Running this code will give us the y-intercept, which is $y = -5$ .

Conclusion

In this article, we have learned how to find the vertex, axis of symmetry, and y-intercept of a quadratic function. We have used the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex, and then used this value to find the y-coordinate of the vertex. We have also used the formula $x = -\frac{b}{2a}$ to find the equation of the axis of symmetry, and finally, we have used the original function to find the y-intercept. By following these steps, we can easily find the vertex, axis of symmetry, and y-intercept of any quadratic function.

Example Problems

Problem 1

Find the vertex, axis of symmetry, and y-intercept of the graph of the function $y = x^2 + 4x + 3$ .

Solution

To find the vertex, we need to plug in the values of $a$ and $b$ into the formula $x = -\frac{b}{2a}$ . In this function, the values of $a$ and $b$ are $1$ and $4$ , respectively.

a = 1
b = 4
x_vertex = -b / (2 * a)
print(x_vertex)

Running this code will give us the x-coordinate of the vertex, which is $x = -2$ .

To find the y-coordinate of the vertex, we need to plug this value into the original function.

y_vertex = (-2)**2 + 4 * (-2) + 3
print(y_vertex)

Running this code will give us the y-coordinate of the vertex, which is $y = -1$ .

The equation of the axis of symmetry is given by $x = -\frac{b}{2a}$ , which is the same formula we used to find the x-coordinate of the vertex.

The y-intercept of this function is the point where the graph of the function intersects the y-axis. It is the value of $y$ when $x = 0$ .

y_intercept = (0)**2 + 4 * (0) + 3
print(y_intercept)

Running this code will give us the y-intercept, which is $y = 3$ .

Problem 2

Find the vertex, axis of symmetry, and y-intercept of the graph of the function $y = -3x^2 + 6x - 2$ .

Solution

To find the vertex, we need to plug in the values of $a$ and $b$ into the formula $x = -\frac{b}{2a}$ . In this function, the values of $a$ and $b$ are $-3$ and $6$ , respectively.

a = -3
b = 6
x_vertex = -b / (2 * a)
print(x_vertex)

Running this code will give us the x-coordinate of the vertex, which is $x = 2$ .

To find the y-coordinate of the vertex, we need to plug this value into the original function.

y_vertex = -3 * (2)**2 + 6 * (2) - 2
print(y_vertex)

Running this code will give us the y-coordinate of the vertex, which is $y = 6$ .

The equation of the axis of symmetry is given by $x = -\frac{b}{2a}$ , which is the same formula we used to find the x-coordinate of the vertex.

The y-intercept of this function is the point where the graph of the function intersects the y-axis. It is the value of $y$ when $x = 0$ .

y_intercept = -3 * (0)**2 + 6 * (0) - 2
print(y_intercept)

Running this code will give us the y-intercept, which is $y = -2$ .

Final Thoughts

Introduction

In our previous article, we learned how to find the vertex, axis of symmetry, and y-intercept of a quadratic function. However, we understand that there may be some questions and doubts that still need to be clarified. In this article, we will address some of the most frequently asked questions about quadratic functions and provide additional examples to help solidify your understanding.

Q1: What is the vertex of a quadratic function?

A1: The vertex of a quadratic function is the highest or lowest point on the graph of the function. It is the point where the function changes from decreasing to increasing or vice versa.

Q2: How do I find the x-coordinate of the vertex?

A2: To find the x-coordinate of the vertex, you need to plug in the values of $a$ and $b$ into the formula $x = -\frac{b}{2a}$ .

Q3: How do I find the y-coordinate of the vertex?

A3: To find the y-coordinate of the vertex, you need to plug the x-coordinate into the original function.

Q4: What is the axis of symmetry?

A4: The axis of symmetry is a vertical line that passes through the vertex of the quadratic function. It is the line that divides the graph of the function into two symmetrical parts.

Q5: How do I find the equation of the axis of symmetry?

A5: To find the equation of the axis of symmetry, you need to plug in the values of $a$ and $b$ into the formula $x = -\frac{b}{2a}$ .

Q6: What is the y-intercept of a quadratic function?

A6: The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. It is the value of $y$ when $x = 0$ .

Q7: How do I find the y-intercept of a quadratic function?

A7: To find the y-intercept of a quadratic function, you need to plug in $x = 0$ into the original function.

Q8: Can I use a calculator to find the vertex, axis of symmetry, and y-intercept of a quadratic function?

A8: Yes, you can use a calculator to find the vertex, axis of symmetry, and y-intercept of a quadratic function. However, it is still important to understand the formulas and concepts behind these calculations.

Q9: What are some real-world applications of quadratic functions?

A9: Quadratic functions have many real-world applications, including modeling the trajectory of a projectile, the spread of a disease, and the motion of an object under the influence of gravity.

Q10: Can I use quadratic functions to model real-world phenomena that are not quadratic in nature?

A10: While quadratic functions can be used to model many real-world phenomena, they may not always be the best choice. In some cases, other types of functions, such as linear or exponential functions, may be more suitable.

Example Problems

Problem 1

Find the vertex, axis of symmetry, and y-intercept of the graph of the function $y = 2x^2 - 4x + 3$ .

Solution

To find the vertex, we need to plug in the values of $a$ and $b$ into the formula $x = -\frac{b}{2a}$ . In this function, the values of $a$ and $b$ are $2$ and $-4$ , respectively.

a = 2
b = -4
x_vertex = -b / (2 * a)
print(x_vertex)

Running this code will give us the x-coordinate of the vertex, which is $x = 1$ .

To find the y-coordinate of the vertex, we need to plug this value into the original function.

y_vertex = 2 * (1)**2 - 4 * (1) + 3
print(y_vertex)

Running this code will give us the y-coordinate of the vertex, which is $y = 1$ .

The equation of the axis of symmetry is given by $x = -\frac{b}{2a}$ , which is the same formula we used to find the x-coordinate of the vertex.

The y-intercept of this function is the point where the graph of the function intersects the y-axis. It is the value of $y$ when $x = 0$ .

y_intercept = 2 * (0)**2 - 4 * (0) + 3
print(y_intercept)

Running this code will give us the y-intercept, which is $y = 3$ .

Problem 2

Find the vertex, axis of symmetry, and y-intercept of the graph of the function $y = -x^2 + 2x - 1$ .

Solution

To find the vertex, we need to plug in the values of $a$ and $b$ into the formula $x = -\frac{b}{2a}$ . In this function, the values of $a$ and $b$ are $-1$ and $2$ , respectively.

a = -1
b = 2
x_vertex = -b / (2 * a)
print(x_vertex)

Running this code will give us the x-coordinate of the vertex, which is $x = -2$ .

To find the y-coordinate of the vertex, we need to plug this value into the original function.

y_vertex = -(-2)**2 + 2 * (-2) - 1
print(y_vertex)

Running this code will give us the y-coordinate of the vertex, which is $y = -7$ .

The equation of the axis of symmetry is given by $x = -\frac{b}{2a}$ , which is the same formula we used to find the x-coordinate of the vertex.

The y-intercept of this function is the point where the graph of the function intersects the y-axis. It is the value of $y$ when $x = 0$ .

y_intercept = -(0)**2 + 2 * (0) - 1
print(y_intercept)

Running this code will give us the y-intercept, which is $y = -1$ .

Conclusion

In this article, we have addressed some of the most frequently asked questions about quadratic functions and provided additional examples to help solidify your understanding. We have learned how to find the vertex, axis of symmetry, and y-intercept of a quadratic function, and how to use these concepts to model real-world phenomena. By following these steps and practicing with example problems, you can become proficient in working with quadratic functions and apply them to a wide range of real-world applications.