Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Let's analyze the function [tex]\( P(x) = x^3 + x^2 - 42x \)[/tex].
1. Finding the [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex].
Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ P(0) = 0^3 + 0^2 - 42 \cdot 0 = 0 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\( 0 \)[/tex].
2. Finding the [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts are the values of [tex]\( x \)[/tex] where the function [tex]\( P(x) = 0 \)[/tex].
We solve the equation:
[tex]\[ x^3 + x^2 - 42x = 0 \][/tex]
Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x(x^2 + x - 42) = 0 \][/tex]
This gives us one [tex]\( x \)[/tex]-intercept at [tex]\( x = 0 \)[/tex].
Now, we solve the quadratic equation [tex]\( x^2 + x - 42 \)[/tex]:
[tex]\[ x^2 + x - 42 = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -42 \)[/tex]:
[tex]\[ x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-42)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 168}}{2} = \frac{-1 \pm \sqrt{169}}{2} = \frac{-1 \pm 13}{2} \][/tex]
Therefore, we get:
[tex]\[ x = \frac{-1 + 13}{2} = 6 \quad \text{and} \quad x = \frac{-1 - 13}{2} = -7 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( -7 \)[/tex], [tex]\( 0 \)[/tex], and [tex]\( 6 \)[/tex].
3. Analyzing the end behavior as [tex]\( x \rightarrow \infty \)[/tex]:
As [tex]\( x \)[/tex] approaches infinity, the term [tex]\( x^3 \)[/tex] will dominate because it has the highest power.
Therefore:
[tex]\[ \lim_ {x \to \infty} (x^3 + x^2 - 42x) = \infty \][/tex]
4. Analyzing the end behavior as [tex]\( x \rightarrow -\infty \)[/tex]:
Similarly, as [tex]\( x \)[/tex] approaches negative infinity, the term [tex]\( x^3 \)[/tex] will still dominate.
Therefore:
[tex]\[ \lim_ {x \to -\infty} (x^3 + x^2 - 42x) = -\infty \][/tex]
Summarizing all the findings:
- The [tex]\( y \)[/tex]-intercept is [tex]\( \boxed{0} \)[/tex].
- The [tex]\( x \)[/tex]-intercepts are [tex]\( \boxed{-7, 0, 6} \)[/tex].
- When [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \boxed{\infty} \)[/tex].
- When [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow \boxed{-\infty} \)[/tex].
1. Finding the [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept of a function is the value of the function when [tex]\( x = 0 \)[/tex].
Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ P(0) = 0^3 + 0^2 - 42 \cdot 0 = 0 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\( 0 \)[/tex].
2. Finding the [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts are the values of [tex]\( x \)[/tex] where the function [tex]\( P(x) = 0 \)[/tex].
We solve the equation:
[tex]\[ x^3 + x^2 - 42x = 0 \][/tex]
Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x(x^2 + x - 42) = 0 \][/tex]
This gives us one [tex]\( x \)[/tex]-intercept at [tex]\( x = 0 \)[/tex].
Now, we solve the quadratic equation [tex]\( x^2 + x - 42 \)[/tex]:
[tex]\[ x^2 + x - 42 = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -42 \)[/tex]:
[tex]\[ x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-42)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 168}}{2} = \frac{-1 \pm \sqrt{169}}{2} = \frac{-1 \pm 13}{2} \][/tex]
Therefore, we get:
[tex]\[ x = \frac{-1 + 13}{2} = 6 \quad \text{and} \quad x = \frac{-1 - 13}{2} = -7 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\( -7 \)[/tex], [tex]\( 0 \)[/tex], and [tex]\( 6 \)[/tex].
3. Analyzing the end behavior as [tex]\( x \rightarrow \infty \)[/tex]:
As [tex]\( x \)[/tex] approaches infinity, the term [tex]\( x^3 \)[/tex] will dominate because it has the highest power.
Therefore:
[tex]\[ \lim_ {x \to \infty} (x^3 + x^2 - 42x) = \infty \][/tex]
4. Analyzing the end behavior as [tex]\( x \rightarrow -\infty \)[/tex]:
Similarly, as [tex]\( x \)[/tex] approaches negative infinity, the term [tex]\( x^3 \)[/tex] will still dominate.
Therefore:
[tex]\[ \lim_ {x \to -\infty} (x^3 + x^2 - 42x) = -\infty \][/tex]
Summarizing all the findings:
- The [tex]\( y \)[/tex]-intercept is [tex]\( \boxed{0} \)[/tex].
- The [tex]\( x \)[/tex]-intercepts are [tex]\( \boxed{-7, 0, 6} \)[/tex].
- When [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \boxed{\infty} \)[/tex].
- When [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow \boxed{-\infty} \)[/tex].
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
