Use The Vertex And Intercepts To Sketch The Graph Of The Quadratic Function. Give The Equation Of The Parabola's Axis Of Symmetry. Use The Graph To Determine The Domain And Range Of The Function.Given Function: F ( X ) = ( X − 4 ) 2 − 9 F(x) = (x-4)^2 - 9 F ( X ) = ( X − 4 ) 2 − 9 - Use The

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how to sketch their graphs is crucial for various applications in science, engineering, and other fields. In this article, we will focus on using the vertex and intercepts to sketch the graph of the quadratic function $f(x) = (x-4)^2 - 9$. We will also determine the equation of the parabola's axis of symmetry and use the graph to find the domain and range of the function.

Understanding the Quadratic Function

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our given function, $f(x) = (x-4)^2 - 9$, we can expand the squared term to get $f(x) = x^2 - 8x + 16 - 9 = x^2 - 8x + 7$.

Finding the Vertex

The vertex of a quadratic function is the maximum or minimum point on the graph, depending on the direction of the parabola. To find the vertex, we can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. In our case, $a = 1$ and $b = -8$, so we can plug these values into the formula to get $x = -\frac{-8}{2(1)} = 4$. This means that the x-coordinate of the vertex is 4.

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 function. We get $f(4) = (4-4)^2 - 9 = 0 - 9 = -9$. Therefore, the vertex of the parabola is at the point (4, -9).

Finding the Intercepts

The x-intercepts of a quadratic function are the points where the graph crosses the x-axis, and the y-intercept is the point where the graph crosses the y-axis. To find the x-intercepts, we can set the function equal to zero and solve for x. We get $(x-4)^2 - 9 = 0$, which simplifies to $(x-4)^2 = 9$. Taking the square root of both sides, we get $x-4 = \pm 3$, which gives us two possible values for x: $x = 4 + 3 = 7$ and $x = 4 - 3 = 1$. Therefore, the x-intercepts are at the points (7, 0) and (1, 0).

Finding the y-Intercept

To find the y-intercept, we can plug in x = 0 into the function. We get $f(0) = (0-4)^2 - 9 = 16 - 9 = 7$. Therefore, the y-intercept is at the point (0, 7).

Sketching the Graph

Now that we have the vertex and intercepts, we can sketch the graph of the quadratic function. The graph will be a parabola that opens upward, since the coefficient of the squared term is positive. The vertex will be at the point (4, -9), and the x-intercepts will be at the points (7, 0) and (1, 0). The y-intercept will be at the point (0, 7).

Determining the Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. To find the equation of the axis of symmetry, we can use the formula $x = -\frac{b}{2a}$, which we used earlier to find the x-coordinate of the vertex. In this case, the equation of the axis of symmetry is $x = 4$.

Determining the Domain and Range

The domain of a quadratic function is the set of all possible input values, and the range is the set of all possible output values. Since the graph of the quadratic function is a parabola that opens upward, the domain will be all real numbers, and the range will be all real numbers greater than or equal to -9.

Conclusion

In this article, we used the vertex and intercepts to sketch the graph of the quadratic function $f(x) = (x-4)^2 - 9$. We also determined the equation of the parabola's axis of symmetry and used the graph to find the domain and range of the function. By understanding how to sketch the graph of a quadratic function, we can gain a deeper understanding of the properties of these functions and how they can be used to model real-world phenomena.

Key Takeaways

• The vertex of a quadratic function is the maximum or minimum point on the graph, depending on the direction of the parabola.
• The x-intercepts of a quadratic function are the points where the graph crosses the x-axis, and the y-intercept is the point where the graph crosses the y-axis.
• The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola.
• The domain of a quadratic function is the set of all possible input values, and the range is the set of all possible output values.

Final Thoughts

Quadratic functions are a fundamental concept in mathematics, and understanding how to sketch their graphs is crucial for various applications in science, engineering, and other fields. By using the vertex and intercepts to sketch the graph of a quadratic function, we can gain a deeper understanding of the properties of these functions and how they can be used to model real-world phenomena.

Introduction

In our previous article, we discussed how to use the vertex and intercepts to sketch the graph of a quadratic function. We also determined the equation of the parabola's axis of symmetry and used the graph to find the domain and range of the function. In this article, we will answer some frequently asked questions about quadratic function graphing.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the maximum or minimum point on the graph, depending on the direction of the parabola. It is the point where the graph changes direction.

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

A: To find the vertex of a quadratic function, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. You can then plug this value into the function to find the y-coordinate of the vertex.

Q: What are the x-intercepts of a quadratic function?

A: The x-intercepts of a quadratic function are the points where the graph crosses the x-axis. They are the values of x where the function equals zero.

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function, you can set the function equal to zero and solve for x. This will give you the values of x where the graph crosses the x-axis.

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

A: The y-intercept of a quadratic function is the point where the graph crosses the y-axis. It is the value of y when x equals zero.

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

A: To find the y-intercept of a quadratic function, you can plug x = 0 into the function. This will give you the value of y when x equals zero.

Q: What is the axis of symmetry of a quadratic function?

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

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

A: To find the axis of symmetry of a quadratic function, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. This will give you the equation of the axis of symmetry.

Q: What is the domain of a quadratic function?

A: The domain of a quadratic function is the set of all possible input values. It is the set of all real numbers.

Q: What is the range of a quadratic function?

A: The range of a quadratic function is the set of all possible output values. It is the set of all real numbers greater than or equal to the minimum value of the function.

Q: How do I determine the domain and range of a quadratic function?

A: To determine the domain and range of a quadratic function, you can use the graph of the function. The domain will be all real numbers, and the range will be all real numbers greater than or equal to the minimum value of the function.

Q: Can I use technology to graph a quadratic function?

A: Yes, you can use technology such as graphing calculators or computer software to graph a quadratic function. This can be a useful tool for visualizing the graph of a quadratic function and determining its properties.

Q: What are some common mistakes to avoid when graphing a quadratic function?

A: Some common mistakes to avoid when graphing a quadratic function include:

• Not using the correct formula for the vertex
• Not plugging the correct value into the function to find the y-coordinate of the vertex
• Not setting the function equal to zero to find the x-intercepts
• Not plugging x = 0 into the function to find the y-intercept
• Not using the correct formula for the axis of symmetry

Conclusion

In this article, we answered some frequently asked questions about quadratic function graphing. We discussed the vertex, x-intercepts, y-intercept, axis of symmetry, domain, and range of a quadratic function. We also provided some tips for avoiding common mistakes when graphing a quadratic function. By understanding these concepts and using the correct formulas and techniques, you can graph a quadratic function and determine its properties with ease.