Find The \[$ X \$\]-intercepts And \[$ Y \$\]-intercepts Of The Function:$\[ G(x) = 3(x-6)x + 2 \\]
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Introduction
In mathematics, the x-intercept and y-intercept of a function are the points where the graph of the function intersects the x-axis and y-axis, respectively. These intercepts are crucial in understanding the behavior of a function and can be used to determine its key characteristics. In this article, we will focus on finding the x- and y-intercepts of a quadratic function, specifically the function g(x) = 3(x-6)x + 2.
What are x- and y-intercepts?
The x-intercept of a function is the point where the graph of the function intersects the x-axis. At this point, the value of y is always 0. Similarly, the y-intercept of a function is the point where the graph of the function intersects the y-axis. At this point, the value of x is always 0.
Finding the x-intercept
To find the x-intercept of a quadratic function, we need to set the value of y to 0 and solve for x. In the case of the function g(x) = 3(x-6)x + 2, we can set y to 0 and solve for x as follows:
g(x) = 3(x-6)x + 2 0 = 3(x-6)x + 2
Step 1: Expand the equation
First, we need to expand the equation by multiplying the terms inside the parentheses.
0 = 3(x^2 - 6x) + 2 0 = 3x^2 - 18x + 2
Step 2: Rearrange the equation
Next, we need to rearrange the equation so that it is in the standard form of a quadratic equation, ax^2 + bx + c = 0.
3x^2 - 18x + 2 = 0
Step 3: Solve for x
Now, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 3, b = -18, and c = 2. Plugging these values into the quadratic formula, we get:
x = (18 ± √((-18)^2 - 4(3)(2))) / (2(3)) x = (18 ± √(324 - 24)) / 6 x = (18 ± √300) / 6 x = (18 ± 10√3) / 6
Step 1: Simplify the solutions
Simplifying the solutions, we get:
x = (18 + 10√3) / 6 or x = (18 - 10√3) / 6
Step 2: Rationalize the denominators
To rationalize the denominators, we can multiply the numerator and denominator by 6.
x = (18 + 10√3) / 6 × (6 / 6) or x = (18 - 10√3) / 6 × (6 / 6) x = (108 + 60√3) / 36 or x = (108 - 60√3) / 36
Step 3: Simplify the solutions
Simplifying the solutions, we get:
x = (108 + 60√3) / 36 or x = (108 - 60√3) / 36
Step 4: Reduce the fractions
To reduce the fractions, we can divide both the numerator and denominator by their greatest common divisor.
x = (3 + 5√3) / 3 or x = (3 - 5√3) / 3
Step 5: Simplify the solutions
Simplifying the solutions, we get:
x = 3 + 5√3 or x = 3 - 5√3
Step 2: Finding the y-intercept
To find the y-intercept of a quadratic function, we need to set the value of x to 0 and solve for y. In the case of the function g(x) = 3(x-6)x + 2, we can set x to 0 and solve for y as follows:
g(x) = 3(x-6)x + 2 g(0) = 3(0-6)(0) + 2 g(0) = 3(-6)(0) + 2 g(0) = 0 + 2 g(0) = 2
Conclusion
In this article, we have found the x- and y-intercepts of the quadratic function g(x) = 3(x-6)x + 2. The x-intercepts are x = 3 + 5√3 and x = 3 - 5√3, and the y-intercept is y = 2. These intercepts are crucial in understanding the behavior of the function and can be used to determine its key characteristics.
Final Answer
The x-intercepts of the function g(x) = 3(x-6)x + 2 are x = 3 + 5√3 and x = 3 - 5√3, and the y-intercept is y = 2.
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Introduction
In our previous article, we discussed how to find the x- and y-intercepts of a quadratic function. In this article, we will answer some frequently asked questions related to finding the x- and y-intercepts of a quadratic function.
Q: What is the x-intercept of a quadratic function?
A: The x-intercept of a quadratic function is the point where the graph of the function intersects the x-axis. At this point, the value of y is always 0.
Q: How do I find the x-intercept of a quadratic function?
A: To find the x-intercept of a quadratic function, you need to set the value of y to 0 and solve for x. You can use the quadratic formula to solve for x.
Q: What is the y-intercept of a quadratic function?
A: The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. At this point, the value of x is always 0.
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 the value of x to 0 and solve for y.
Q: Can I use a graphing calculator to find the x- and y-intercepts of a quadratic function?
A: Yes, you can use a graphing calculator to find the x- and y-intercepts of a quadratic function. Graphing calculators can help you visualize the graph of the function and find the x- and y-intercepts.
Q: What if the quadratic function has no real roots?
A: If the quadratic function has no real roots, it means that the function does not intersect the x-axis. In this case, the x-intercept is undefined.
Q: Can I find the x- and y-intercepts of a quadratic function with complex roots?
A: Yes, you can find the x- and y-intercepts of a quadratic function with complex roots. However, the x-intercepts will be complex numbers.
Q: How do I find the x- and y-intercepts of a quadratic function with complex roots?
A: To find the x- and y-intercepts of a quadratic function with complex roots, you need to use the quadratic formula and simplify the expression. You may need to use complex numbers to represent the roots.
Q: Can I use the quadratic formula to find the x- and y-intercepts of a quadratic function with complex roots?
A: Yes, you can use the quadratic formula to find the x- and y-intercepts of a quadratic function with complex roots. However, you need to be careful when simplifying the expression and using complex numbers.
Q: What if the quadratic function has a repeated root?
A: If the quadratic function has a repeated root, it means that the function intersects the x-axis at a single point. In this case, the x-intercept is a single point.
Q: Can I find the x- and y-intercepts of a quadratic function with a repeated root?
A: Yes, you can find the x- and y-intercepts of a quadratic function with a repeated root. However, the x-intercept will be a single point.
Q: How do I find the x- and y-intercepts of a quadratic function with a repeated root?
A: To find the x- and y-intercepts of a quadratic function with a repeated root, you need to use the quadratic formula and simplify the expression. You may need to use the fact that the root is repeated to find the x-intercept.
Conclusion
In this article, we have answered some frequently asked questions related to finding the x- and y-intercepts of a quadratic function. We hope that this article has been helpful in clarifying any doubts you may have had about finding the x- and y-intercepts of a quadratic function.
Final Answer
The x-intercepts of a quadratic function are the points where the graph of the function intersects the x-axis. The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. You can use the quadratic formula to find the x- and y-intercepts of a quadratic function.