Sketch the graph of the polynomial function $$ p(x) = -x^5+4x^4+x^3-15x^2+12x+4 $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -x^5+4x^4+x^3-15x^2+12x+4 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = -0.2533 & x_2 = -1.9295 & x_3 = 3.2888 \end{matrix} $$

(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^5+4x^4+x^3-15x^2+12x+4 } $, so:

$$ \text{Y inercept} = p(0) = 4 $$

STEP 3:Find the end behavior

The end behavior of a polynomial is the same as the end behavior of a leading term.

$$ \lim_{x \to -\infty} \left( -x^5+4x^4+x^3-15x^2+12x+4 \right) = \lim_{x \to -\infty} -x^5 = \color{blue}{ \infty } $$

The graph starts in the upper-left corner.

$$ \lim_{x \to \infty} \left( -x^5+4x^4+x^3-15x^2+12x+4 \right) = \lim_{x \to \infty} -x^5 = \color{blue}{ -\infty } $$

The graph ends in the lower-right corner.

STEP 4:Find the turning points

To determine the turning points, we need to find the first derivative of $ p(x) $:

$$ p^{\prime} (x) = -5x^4+16x^3+3x^2-30x+12 $$

The x coordinate of the turning points are located at the zeros of the first derivative

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.4689 & x_2 = -1.3698 & x_3 = 1.3665 & x_4 = 2.7343 \end{matrix} $$

(cleck here to see a explanation of how to solve the equation)

To find the y coordinates, substitute the above values into $ p(x) $

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.4689 } \Rightarrow p\left(0.4689\right) = \color{orangered}{ 6.6026 }\\[1 em] \text{for } ~ x & = \color{blue}{ -1.3698 } \Rightarrow p\left(-1.3698\right) = \color{orangered}{ -24.2477 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.3665 } \Rightarrow p\left(1.3665\right) = \color{orangered}{ 4.1226 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.7343 } \Rightarrow p\left(2.7343\right) = \color{orangered}{ 15.8566 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.4689, 6.6026 \right) & \left( -1.3698, -24.2477 \right) & \left( 1.3665, 4.1226 \right) & \left( 2.7343, 15.8566 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -20x^3+48x^2+6x-30 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 0.9069 & x_2 = -0.7405 & x_3 = 2.2337 \end{matrix} $$

Substitute the x values into $ p(x) $ to get y coordinates

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.9069 } \Rightarrow p\left(0.9069\right) = \color{orangered}{ 5.3842 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.7405 } \Rightarrow p\left(-0.7405\right) = \color{orangered}{ -12.0923 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.2337 } \Rightarrow p\left(2.2337\right) = \color{orangered}{ 11.0782 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 0.9069, 5.3842 \right) & \left( -0.7405, -12.0923 \right) & \left( 2.2337, 11.0782 \right)\end{matrix} $$

Graphing Polynomials