Sketch the graph of the polynomial function $$ p(x) = -\frac{7}{4}x^4+\frac{7}{3}x^3+\frac{9}{2}x^2-x $$

Tap the blue points to see coordinates.

STEP 1:Find the x-intercepts

To find the x-intercepts solve, the equation $ \color{blue}{ -\frac{7}{4}x^4+\frac{7}{3}x^3+\frac{9}{2}x^2-x = 0 } $

The solutions of this equation are:

$$ \begin{matrix}x_1 = 0 & x_2 = 0.204 & x_3 = -1.2018 & x_4 = 2.3312 \end{matrix} $$

(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) = -\frac{7}{4}x^4+\frac{7}{3}x^3+\frac{9}{2}x^2-x } $, so:

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

STEP 3:Find the end behavior

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

$$ \lim_{x \to -\infty} \left( -\frac{7}{4}x^4+\frac{7}{3}x^3+\frac{9}{2}x^2-x \right) = \lim_{x \to -\infty} -\frac{7}{4}x^4 = \color{blue}{ -\infty } $$

The graph starts in the lower-left corner.

$$ \lim_{x \to \infty} \left( -\frac{7}{4}x^4+\frac{7}{3}x^3+\frac{9}{2}x^2-x \right) = \lim_{x \to \infty} -\frac{7}{4}x^4 = \color{blue}{ -\infty } $$

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) $:

$$ p^{\prime} (x) = -7x^3+7x^2+9x-1 $$

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

$$ p^{\prime} (x) = 0 $$ $$ \begin{matrix}x_1 = 0.1036 & x_2 = -0.8086 & x_3 = 1.705 \end{matrix} $$

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

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 0.1036 } \Rightarrow p\left(0.1036\right) = \color{orangered}{ -0.0529 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.8086 } \Rightarrow p\left(-0.8086\right) = \color{orangered}{ 1.7691 }\\[1 em] \text{for } ~ x & = \color{blue}{ 1.705 } \Rightarrow p\left(1.705\right) = \color{orangered}{ 8.1528 }\end{aligned} $$

So the turning points are:

$$ \begin{matrix} \left( 0.1036, -0.0529 \right) & \left( -0.8086, 1.7691 \right) & \left( 1.705, 8.1528 \right)\end{matrix} $$

STEP 5:Find the inflection points

The inflection points are located at zeroes of second derivative. The second derivative is $ p^{\prime \prime} (x) = -21x^2+14x+9 $.

The zeros of second derivative are

$$ \begin{matrix}x_1 = 1.068 & x_2 = -0.4013 \end{matrix} $$

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

$$ \begin{aligned} \text{for } ~ x & = \color{blue}{ 1.068 } \Rightarrow p\left(1.068\right) = \color{orangered}{ 4.6302 }\\[1 em] \text{for } ~ x & = \color{blue}{ -0.4013 } \Rightarrow p\left(-0.4013\right) = \color{orangered}{ 0.9298 }\end{aligned} $$

So the inflection points are:

$$ \begin{matrix} \left( 1.068, 4.6302 \right) & \left( -0.4013, 0.9298 \right)\end{matrix} $$

Graphing Polynomials