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

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

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

The solutions of this equation are:

$$ \begin{matrix}x_1 = 1.7248 & x_2 = 3.1275 \end{matrix} $$

(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^4-4x^3+2x^2+x+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.

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

The graph starts in the upper-left corner.

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

The graph ends in the upper-right corner.

STEP 4:Find the turning points

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

$$ p^{\prime} (x) = 4x^3-12x^2+4x+1 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = -0.1644 & x_2 = 0.5907 & x_3 = 2.5737 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.1644 } \Rightarrow p\left(-0.1644\right) = \color{orangered}{ 3.9082 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.5907 } \Rightarrow p\left(0.5907\right) = \color{orangered}{ 4.5859 }\\[1 em] \text{for } ~ x & = \color{blue}{ 2.5737 } \Rightarrow p\left(2.5737\right) = \color{orangered}{ -4.494 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( -0.1644, 3.9082 \right) & \left( 0.5907, 4.5859 \right) & \left( 2.5737, -4.494 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = 12x^2-24x+4 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 1.8165 & x_2 = 0.1835 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.8165 } \Rightarrow p\left(1.8165\right) = \color{orangered}{ -0.6717 }\\[1 em] \text{for } ~ x & = \color{blue}{ 0.1835 } \Rightarrow p\left(0.1835\right) = \color{orangered}{ 4.2273 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 1.8165, -0.6717 \right) & \left( 0.1835, 4.2273 \right)\end{matrix} $$

Graphing Polynomials