$x$-intercepts: Domain: $(-\infty, \infty)$Range: $-9 \ \textless \ Y \ \textless \ 6 \ \textless \ F(x)$Example 2: $f(x) = (x+3)^2 - 4$Does It Open Up Or Down?Vertex: Axis Of Symmetry:

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Understanding the Basics of $x$-intercepts

In mathematics, the $x$-intercept of a function is the point at which the graph of the function crosses the $x$-axis. This occurs when the value of $y$ is equal to zero. The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values. In this case, the domain is $(-\infty, \infty)$, indicating that the function is defined for all real numbers, and the range is $-9 \ \textless \ y \ \textless \ 6$, indicating that the output values of the function lie between -9 and 6.

Example 2: $f(x) = (x+3)^2 - 4$

To determine whether the function opens up or down, we need to examine the coefficient of the squared term. In this case, the coefficient is 1, which is positive. This indicates that the function will open up.

Determining the Vertex and Axis of Symmetry

The vertex of a quadratic function is the point at which the function reaches its maximum or minimum value. To find the vertex, we need to use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the squared term and the linear term, respectively. In this case, $a = 1$ and $b = 6$, so the vertex is located at $x = -\frac{6}{2(1)} = -3$.

The axis of symmetry is a vertical line that passes through the vertex and is perpendicular to the $x$-axis. Since the vertex is located at $x = -3$, the axis of symmetry is the line $x = -3$.

Discussion Category: Mathematics

In mathematics, the study of $x$-intercepts and the properties of quadratic functions is a fundamental concept. Understanding how to determine the vertex and axis of symmetry of a quadratic function is crucial for solving problems in algebra and calculus.

Key Takeaways

  • The $x$-intercept of a function is the point at which the graph of the function crosses the $x$-axis.
  • The domain of a function is the set of all possible input values for which the function is defined.
  • The range of a function is the set of all possible output values.
  • A quadratic function with a positive coefficient of the squared term will open up.
  • The vertex of a quadratic function is the point at which the function reaches its maximum or minimum value.
  • The axis of symmetry is a vertical line that passes through the vertex and is perpendicular to the $x$-axis.

Example Problems

  1. Find the $x$-intercept of the function $f(x) = (x-2)^2 + 1$.
  2. Determine the domain and range of the function $f(x) = \frac{1}{x-2}$.
  3. Find the vertex and axis of symmetry of the function $f(x) = (x+2)^2 - 3$.

Solutions to Example Problems

  1. To find the $x$-intercept of the function $f(x) = (x-2)^2 + 1$, we need to set the value of $y$ equal to zero and solve for $x$. This gives us the equation $(x-2)^2 + 1 = 0$, which has no real solutions. Therefore, the function has no $x$-intercept.
  2. The domain of the function $f(x) = \frac{1}{x-2}$ is all real numbers except $x = 2$, since the function is undefined when the denominator is equal to zero. The range of the function is all real numbers except zero, since the function is never equal to zero.
  3. To find the vertex and axis of symmetry of the function $f(x) = (x+2)^2 - 3$, we need to use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the squared term and the linear term, respectively. In this case, $a = 1$ and $b = 4$, so the vertex is located at $x = -\frac{4}{2(1)} = -2$. The axis of symmetry is the line $x = -2$.

Conclusion

In conclusion, the study of $x$-intercepts and the properties of quadratic functions is a fundamental concept in mathematics. Understanding how to determine the vertex and axis of symmetry of a quadratic function is crucial for solving problems in algebra and calculus. By following the steps outlined in this article, you can gain a deeper understanding of these concepts and improve your problem-solving skills.

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Q&A: $x$-intercepts, Domain, and Range

Q: What is the $x$-intercept of a function? A: The $x$-intercept of a function is the point at which the graph of the function crosses the $x$-axis. This occurs when the value of $y$ is equal to zero.

Q: What is the domain of a function? A: The domain of a function is the set of all possible input values for which the function is defined.

Q: What is the range of a function? A: The range of a function is the set of all possible output values.

Q: How do I determine whether a quadratic function opens up or down? A: To determine whether a quadratic function opens up or down, we need to examine the coefficient of the squared term. If the coefficient is positive, the function will open up. If the coefficient is negative, the function will open down.

Q: How do I find the vertex of a quadratic function? A: To find the vertex of a quadratic function, we need to use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the squared term and the linear term, respectively.

Q: What is the axis of symmetry of a quadratic function? A: The axis of symmetry is a vertical line that passes through the vertex and is perpendicular to the $x$-axis.

Q: How do I determine the domain and range of a function? A: To determine the domain and range of a function, we need to examine the function and identify any restrictions on the input values and output values.

Q: What is the difference between the domain and range of a function? A: The domain of a function is the set of all possible input values, while the range of a function is the set of all possible output values.

Q: Can a function have multiple $x$-intercepts? A: Yes, a function can have multiple $x$-intercepts.

Q: Can a function have multiple vertices? A: No, a function can only have one vertex.

Q: Can a function have multiple axes of symmetry? A: No, a function can only have one axis of symmetry.

Example Problems with Solutions

  1. Find the $x$-intercept of the function $f(x) = (x-2)^2 + 1$.
    • Solution: The function has no $x$-intercept, since the equation $(x-2)^2 + 1 = 0$ has no real solutions.
  2. Determine the domain and range of the function $f(x) = \frac{1}{x-2}$.
    • Solution: The domain is all real numbers except $x = 2$, and the range is all real numbers except zero.
  3. Find the vertex and axis of symmetry of the function $f(x) = (x+2)^2 - 3$.
    • Solution: The vertex is located at $x = -2$, and the axis of symmetry is the line $x = -2$.

Conclusion

In conclusion, the study of $x$-intercepts, domain, and range is a fundamental concept in mathematics. Understanding how to determine the vertex and axis of symmetry of a quadratic function is crucial for solving problems in algebra and calculus. By following the steps outlined in this article, you can gain a deeper understanding of these concepts and improve your problem-solving skills.