Sketch the graph of the polynomial function $$ p(x) = x^5-4x^4-60x^3+226x^2+539x-1470 $$

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^4-60x^3+226x^2+539x-1470 = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 2 & x_2 = 5 & x_3 = 7 & x_4 = -3 & x_5 = -7 \end{matrix} $$

(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^4-60x^3+226x^2+539x-1470 } $, so:

$$ \text{Y inercept} = p(0) = -1470 $$

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( x^5-4x^4-60x^3+226x^2+539x-1470 \right) = \lim_{x \to -\infty} x^5 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( x^5-4x^4-60x^3+226x^2+539x-1470 \right) = \lim_{x \to \infty} x^5 = \color{blue}{ \infty } $$

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) $:

$$ p^{\prime} (x) = 5x^4-16x^3-180x^2+452x+539 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = -0.9019 & x_2 = -5.5641 & x_3 = 3.4635 & x_4 = 6.2025 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ -0.9019 } \Rightarrow p\left(-0.9019\right) = \color{orangered}{ -1731.5161 }\\[1 em] \text{for } ~ x & = \color{blue}{ -5.5641 } \Rightarrow p\left(-5.5641\right) = \color{orangered}{ 3696.4114 }\\[1 em] \text{for } ~ x & = \color{blue}{ 3.4635 } \Rightarrow p\left(3.4635\right) = \color{orangered}{ 537.8283 }\\[1 em] \text{for } ~ x & = \color{blue}{ 6.2025 } \Rightarrow p\left(6.2025\right) = \color{orangered}{ -489.6497 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( -0.9019, -1731.5161 \right) & \left( -5.5641, 3696.4114 \right) & \left( 3.4635, 537.8283 \right) & \left( 6.2025, -489.6497 \right)\end{matrix} $$

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^2-360x+452 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 1.1626 & x_2 = -3.8334 & x_3 = 5.0708 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.1626 } \Rightarrow p\left(1.1626\right) = \color{orangered}{ -637.3327 }\\[1 em] \text{for } ~ x & = \color{blue}{ -3.8334 } \Rightarrow p\left(-3.8334\right) = \color{orangered}{ 1473.2724 }\\[1 em] \text{for } ~ x & = \color{blue}{ 5.0708 } \Rightarrow p\left(5.0708\right) = \color{orangered}{ -40.8615 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 1.1626, -637.3327 \right) & \left( -3.8334, 1473.2724 \right) & \left( 5.0708, -40.8615 \right)\end{matrix} $$

Graphing Polynomials