Consider The Function Below:$\[ F(x) = X^2 - 6x + 5 \\]Use The Drawing Tool(s) To Form The Correct Answer On The Provided Coordinate Plane. Plot The Following For The Function:- \[$ X \$\]-intercept(s)- \[$ Y \$\]-intercept-

The given function is $f(x) = x^2 - 6x + 5$. To understand the behavior of this function, we need to analyze its graph. The graph of a quadratic function is a parabola, which can be either upward-facing or downward-facing. In this case, we need to determine the direction of the parabola and find the $x$-intercept(s) and $y$-intercept.

Finding the $x$-Intercept(s)

The $x$-intercept(s) of a function are the points where the graph of the function crosses the $x$-axis. To find the $x$-intercept(s), we need to set $y$ equal to zero and solve for $x$. In this case, we have:

$$f(x) = x^2 - 6x + 5 = 0$$

We can solve this quadratic equation using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a = 1$, $b = -6$, and $c = 5$. Plugging in these values, we get:

$$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(5)}}{2(1)}$$

$$x = \frac{6 \pm \sqrt{36 - 20}}{2}$$

$$x = \frac{6 \pm \sqrt{16}}{2}$$

$$x = \frac{6 \pm 4}{2}$$

Therefore, the $x$-intercept(s) are:

$$x = \frac{6 + 4}{2} = 5$$

$$x = \frac{6 - 4}{2} = 1$$

Finding the $y$-Intercept

The $y$-intercept of a function is the point where the graph of the function crosses the $y$-axis. To find the $y$-intercept, we need to set $x$ equal to zero and solve for $y$. In this case, we have:

$$f(x) = x^2 - 6x + 5$$

Setting $x$ equal to zero, we get:

$$f(0) = 0^2 - 6(0) + 5$$

$$f(0) = 5$$

Therefore, the $y$-intercept is $(0, 5)$.

Plotting the Graph

To plot the graph of the function, we need to use the $x$-intercept(s) and $y$-intercept. We can start by plotting the $y$-intercept at $(0, 5)$. Then, we can plot the $x$-intercept(s) at $(5, 0)$ and $(1, 0)$.

Using the drawing tool(s), we can plot the graph of the function. The graph of the function is a parabola that opens upward. The $x$-intercept(s) are at $(5, 0)$ and $(1, 0)$, and the $y$-intercept is at $(0, 5)$.

Conclusion

In this article, we analyzed the function $f(x) = x^2 - 6x + 5$ and plotted its graph on the provided coordinate plane. We found the $x$-intercept(s) and $y$-intercept and used the drawing tool(s) to plot the graph of the function. The graph of the function is a parabola that opens upward, and the $x$-intercept(s) are at $(5, 0)$ and $(1, 0)$, and the $y$-intercept is at $(0, 5)$.

Key Takeaways

• The function $f(x) = x^2 - 6x + 5$ is a quadratic function that can be analyzed using the quadratic formula.
• The $x$-intercept(s) of a function are the points where the graph of the function crosses the $x$-axis.
• The $y$-intercept of a function is the point where the graph of the function crosses the $y$-axis.
• The graph of a quadratic function is a parabola that can be either upward-facing or downward-facing.

Further Reading

For further reading on quadratic functions and their graphs, we recommend the following resources:

• Khan Academy: Quadratic Functions
• Math Is Fun: Quadratic Functions
• Wolfram MathWorld: Quadratic Function

References

• [1] Khan Academy. Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-equations
• [2] Math Is Fun. Quadratic Functions. Retrieved from https://www.mathisfun.com/algebra/quadratic-functions.html
• [3] Wolfram MathWorld. Quadratic Function. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html
  Quadratic Function Q&A =========================

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic function is:

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

where a, b, and c are constants.

Q: What is the graph of a quadratic function?

A: The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola can open upward or downward, depending on the value of the coefficient a.

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

A: To find the x-intercepts of a quadratic function, you need to set the function equal to zero and solve for x. This can be done using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

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

A: To find the y-intercept of a quadratic function, you need to set x equal to zero and solve for y. This can be done by plugging x = 0 into the function:

$$f(0) = a(0)^2 + b(0) + c$$

$$f(0) = c$$

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the lowest or highest point on the graph of the function. It is the point where the parabola changes direction. The vertex can be found using the formula:

$$x = -\frac{b}{2a}$$

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you need to look at the coefficient a. If a is positive, the parabola opens upward. If a is negative, the parabola opens downward.

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 parabola 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 need to use the formula:

$$x = -\frac{b}{2a}$$

Q: What is the domain and range of a quadratic function?

A: The domain of a quadratic function is all real numbers, while the range is all real numbers greater than or equal to the minimum value of the function.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to plot the x-intercepts, y-intercept, and vertex. You can use a graphing calculator or software to help you graph the function.

Q: What are some common applications of quadratic functions?

A: Quadratic functions have many real-world applications, including:

• Modeling the trajectory of a projectile
• Finding the maximum or minimum value of a function
• Determining the shape of a parabola
• Solving optimization problems

Q: How do I solve quadratic equations?

A: Quadratic equations can be solved using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: What is the difference between a quadratic function and a quadratic equation?

A: A quadratic function is a polynomial function of degree two, while a quadratic equation is an equation that can be written in the form of a quadratic function.

Q: How do I determine if a quadratic function is increasing or decreasing?

A: To determine if a quadratic function is increasing or decreasing, you need to look at the coefficient a. If a is positive, the function is increasing. If a is negative, the function is decreasing.

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

A: The vertex of a quadratic function is the lowest or highest point on the graph of the function. It is the point where the parabola changes direction.

Q: How do I find the equation of a quadratic function given its graph?

A: To find the equation of a quadratic function given its graph, you need to use the x-intercepts, y-intercept, and vertex to determine the values of a, b, and c.

Q: What are some common mistakes to avoid when working with quadratic functions?

A: Some common mistakes to avoid when working with quadratic functions include:

• Not using the correct formula for the axis of symmetry
• Not using the correct formula for the vertex
• Not checking the domain and range of the function
• Not using the correct method to solve quadratic equations