Tap the blue points to see coordinates.
STEP 1:Find the x-intercepts
To find the x-intercepts solve, the equation $ \color{blue}{ x^6-x^5+2x^2+3x-17 = 0 } $
The solutions of this equation are:
$$ \begin{matrix}x_1 = -1.472 & x_2 = 1.6194 \end{matrix} $$(you can use the step-by-step polynomial equation solver to see a detailed explanation of how to solve the equation)
STEP 2:Find the y-intercepts
To find the y-intercepts, substitute $ x = 0 $ into $ \color{blue}{ p(x) = x^6-x^5+2x^2+3x-17 } $, so:
$$ \text{Y inercept} = p(0) = -17 $$STEP 3:Find the end behavior
The end behavior of a polynomial is the same as the end behavior of a leading term.
$$ \lim_{x \to -\infty} \left( x^6-x^5+2x^2+3x-17 \right) = \lim_{x \to -\infty} x^6 = \color{blue}{ \infty } $$The graph starts in the upper-left corner.
$$ \lim_{x \to \infty} \left( x^6-x^5+2x^2+3x-17 \right) = \lim_{x \to \infty} x^6 = \color{blue}{ \infty } $$The graph ends in the upper-right corner.
STEP 4:Find the turning points
To determine the turning point, we need to find the first derivative of $ p(x) $:
$$ p^{\prime} (x) = 6x^5-5x^4+4x+3 $$The x coordinate of the turning point is located at the zeros of the first derivative
$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x = -0.554 \end{matrix} $$(cleck here to see a explanation of how to solve the equation)
To find the y coordinate, substitute the above value into $ p(x) $
$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.554 } \Rightarrow p\left(-0.554\right) = \color{orangered}{ -17.9671 }\end{aligned} $$So the turning point is:
$$ \begin{matrix} \left( -0.554, -17.9671 \right)\end{matrix} $$STEP 5:Find the inflection points
The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 30x^4-20x^3+4 $.
Since above equation has no solutions we conclude that
polynomial has no inflection points.