Given the function $P(x)=(x-1)^{2}(x-8)$, find its $y$-intercept is its $x$-intercepts are $x_{1}=$ and $x_{2}=$ with $x_{1}
Solution
Step 1 ：Given the function \(P(x)=(x-1)^{2}(x-8)\), we need to find its y-intercept and x-intercepts.
Step 2 ：The y-intercept of a function is the point where the graph of the function intersects the y-axis. This occurs when x = 0. So, to find the y-intercept of the function P(x), we substitute x = 0 into the function and calculate the resulting y value. \(P(0) = (0-1)^{2}(0-8) = -8\). So, the y-intercept is \(y=-8\).
Step 3 ：The x-intercepts of a function are the points where the graph of the function intersects the x-axis. This occurs when y = 0. So, to find the x-intercepts of the function P(x), we set the function equal to zero and solve for x. \((x-1)^{2}(x-8) = 0\). The solutions to this equation are \(x_{1}=1\) and \(x_{2}=8\).
Step 4 ：As x approaches infinity, the value of y will depend on the highest power of x in the function. In this case, the highest power of x is 3 (from the term (x-1)^2 * (x-8)), so as x approaches infinity, y will also approach infinity.
Step 5 ：As x approaches negative infinity, the value of y will also depend on the highest power of x in the function. Since the highest power of x is 3 and it is an odd number, as x approaches negative infinity, y will approach negative infinity.
Step 6 ：\(\boxed{\text{Final Answer: The y-intercept of the function } P(x)=(x-1)^{2}(x-8) \text{ is } y=-8. \text{ The x-intercepts are } x_{1}=1 \text{ and } x_{2}=8 \text{ with } x_{1}
From Solvely APP
Solution
Step 1 ：Given the function \(P(x)=(x-1)^{2}(x-8)\), we need to find its y-intercept and x-intercepts.
Step 2 ：The y-intercept of a function is the point where the graph of the function intersects the y-axis. This occurs when x = 0. So, to find the y-intercept of the function P(x), we substitute x = 0 into the function and calculate the resulting y value. \(P(0) = (0-1)^{2}(0-8) = -8\). So, the y-intercept is \(y=-8\).
Step 3 ：The x-intercepts of a function are the points where the graph of the function intersects the x-axis. This occurs when y = 0. So, to find the x-intercepts of the function P(x), we set the function equal to zero and solve for x. \((x-1)^{2}(x-8) = 0\). The solutions to this equation are \(x_{1}=1\) and \(x_{2}=8\).
Step 4 ：As x approaches infinity, the value of y will depend on the highest power of x in the function. In this case, the highest power of x is 3 (from the term (x-1)^2 * (x-8)), so as x approaches infinity, y will also approach infinity.
Step 5 ：As x approaches negative infinity, the value of y will also depend on the highest power of x in the function. Since the highest power of x is 3 and it is an odd number, as x approaches negative infinity, y will approach negative infinity.
Step 6 ：\(\boxed{\text{Final Answer: The y-intercept of the function } P(x)=(x-1)^{2}(x-8) \text{ is } y=-8. \text{ The x-intercepts are } x_{1}=1 \text{ and } x_{2}=8 \text{ with } x_{1}
