# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/polynomial/polynomial-graph Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/polynomial/polynomial-graph/en.mdx Learn polynomial graphing with point plotting, end behavior analysis, and visual patterns. Learn to sketch linear, quadratic, and cubic graphs one step at a time. --- ## Graphing Polynomial Functions The graph of a polynomial function provides a visual representation of how the function's value changes as the input value $$x$$ changes. The shape of this graph can vary greatly depending on the degree and coefficients of the function. Visible text: The graph of a polynomial function provides a visual representation of how the function's value changes as the input value changes. The shape of this graph can vary greatly depending on the degree and coefficients of the function. ## Point Plotting Method The most fundamental way to draw a graph is by determining several $$(x, y)$$ point pairs that satisfy the function, then connecting them with a smooth curve. Visible text: The most fundamental way to draw a graph is by determining several point pairs that satisfy the function, then connecting them with a smooth curve. **Steps:** 1. Choose several different values for $$x$$. 2. Calculate the value $$y = P(x)$$ for each chosen $$x$$ value. 3. Create a table of $$(x, y)$$ value pairs. 4. Plot these points on the coordinate plane. 5. Connect the points with a smooth and continuous curve. Visible text: 1. Choose several different values for . 2. Calculate the value for each chosen value. 3. Create a table of value pairs. 4. Plot these points on the coordinate plane. 5. Connect the points with a smooth and continuous curve. ### Linear Function Degree One Graph the function $$f(x) = 2x + 5$$. Visible text: Graph the function . We choose several $$x$$ values and calculate $$y$$: Visible text: We choose several values and calculate : | $$x$$ | $$y = f(x)$$ | $$(x, y)$$ | | :---------------------- | :----------------------------- | :----------------------------- | | $$-3$$ | $$-1$$ | $$(-3, -1)$$ | | $$-2$$ | $$1$$ | $$(-2, 1)$$ | | $$-1$$ | $$3$$ | $$(-1, 3)$$ | | $$0$$ | $$5$$ | $$(0, 5)$$ | | $$1$$ | $$7$$ | $$(1, 7)$$ | | $$2$$ | $$9$$ | $$(2, 9)$$ | | $$3$$ | $$11$$ | $$(3, 11)$$ | Visible text: | | | | | :---------------------- | :----------------------------- | :----------------------------- | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Plot the points and connect them: Component: LineEquation Props: - title: Graph of $$f(x) = 2x + 5$$ Visible text: Graph of - description: Graph of a linear function with degree $$1$$. Visible text: Graph of a linear function with degree . - showZAxis: false - cameraPosition: [0, 0, 15] - data: [ { points: Array.from({ length: 7 }, (_, i) => { const x = -3 + i; return { x, y: 2 * x + 5, z: 0 }; }), color: getColor("YELLOW"), }, ] ### Quadratic Function Degree Two Graph the function $$g(x) = x^2 - 2x - 3$$. Visible text: Graph the function . Table of values: | $$x$$ | $$y = g(x)$$ | $$(x, y)$$ | | :---------------------- | :----------------------------- | :---------------------------- | | $$-2$$ | $$5$$ | $$(-2, 5)$$ | | $$-1$$ | $$0$$ | $$(-1, 0)$$ | | $$0$$ | $$-3$$ | $$(0, -3)$$ | | $$1$$ | $$-4$$ | $$(1, -4)$$ | | $$2$$ | $$-3$$ | $$(2, -3)$$ | | $$3$$ | $$0$$ | $$(3, 0)$$ | | $$4$$ | $$5$$ | $$(4, 5)$$ | Visible text: | | | | | :---------------------- | :----------------------------- | :---------------------------- | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Plot the points and connect with a smooth curve (parabola): Component: LineEquation Props: - title: Graph of $$g(x) = x^2 - 2x - 3$$ Visible text: Graph of - description: Graph of a quadratic function with degree $$2$$. Visible text: Graph of a quadratic function with degree . - showZAxis: false - cameraPosition: [0, 0, 10] - data: [ { points: Array.from({ length: 7 }, (_, i) => { const x = -2 + i; return { x, y: x * x - 2 * x - 3, z: 0 }; }), color: getColor("CYAN"), }, ] ### Cubic Function Degree Three Graph the function $$h(x) = x^3 + 3x^2 - 4$$. Visible text: Graph the function . Table of values: | $$x$$ | $$y = h(x)$$ | $$(x, y)$$ | | :---------------------- | :----------------------------- | :------------------------------ | | $$-4$$ | $$-20$$ | $$(-4, -20)$$ | | $$-3$$ | $$-4$$ | $$(-3, -4)$$ | | $$-2$$ | $$0$$ | $$(-2, 0)$$ | | $$-1$$ | $$-2$$ | $$(-1, -2)$$ | | $$0$$ | $$-4$$ | $$(0, -4)$$ | | $$1$$ | $$0$$ | $$(1, 0)$$ | | $$2$$ | $$16$$ | $$(2, 16)$$ | Visible text: | | | | | :---------------------- | :----------------------------- | :------------------------------ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Plot the points and connect with a smooth curve: Component: LineEquation Props: - title: Graph of $$h(x) = x^3 + 3x^2 - 4$$ Visible text: Graph of - description: Graph of a cubic function with degree $$3$$. Visible text: Graph of a cubic function with degree . - showZAxis: false - cameraPosition: [0, 0, 12] - data: [ { points: Array.from({ length: 61 }, (_, i) => { // Increased points for smoothness const x = -4 + i * 0.1; return { x, y: x * x * x + 3 * x * x - 4, z: 0 }; }), color: getColor("VIOLET"), showPoints: false, }, ] ## General Characteristics of Polynomial Graphs Graphs of polynomial functions are always **smooth** (no sharp corners) and **continuous** (no jumps or breaks). Their general shape is heavily influenced by the **degree** of the polynomial. - **Degree** $$0$$: $$P(x) = c$$. The graph is a horizontal line. - **Degree** $$1$$: $$P(x) = ax + b$$. The graph is a straight (slanted) line. - **Degree** $$2$$: $$P(x) = ax^2 + bx + c$$. The graph is a parabola. - **Degree** $$3$$: $$P(x) = ax^3 + \dots$$. The graph has a shape like the letter 'S' or a reverse 'S', and can have up to two 'peaks' or 'valleys'. - **Degree** $$4$$: The graph can have up to three 'peaks' or 'valleys'. - **Degree** $$5$$: The graph can have up to four 'peaks' or 'valleys'. Visible text: - **Degree** : . The graph is a horizontal line. - **Degree** : . The graph is a straight (slanted) line. - **Degree** : . The graph is a parabola. - **Degree** : . The graph has a shape like the letter 'S' or a reverse 'S', and can have up to two 'peaks' or 'valleys'. - **Degree** : The graph can have up to three 'peaks' or 'valleys'. - **Degree** : The graph can have up to four 'peaks' or 'valleys'. In general, the graph of a polynomial function of degree $$n$$ can intersect the $$x$$-axis **at most** $$n\text{ times}$$ and has **at most** $$n-1$$ turning points (peaks or valleys). Visible text: In general, the graph of a polynomial function of degree can intersect the -axis **at most** and has **at most** turning points (peaks or valleys). ## End Behavior One important characteristic of polynomial graphs is their **end behavior**, which describes the direction of the graph as $$x$$ approaches positive infinity ($$x \to \infty$$) or negative infinity ($$x \to -\infty$$). Visible text: One important characteristic of polynomial graphs is their **end behavior**, which describes the direction of the graph as approaches positive infinity () or negative infinity (). The end behavior is determined **solely** by the **leading term** $$a_n x^n$$: Visible text: The end behavior is determined **solely** by the **leading term** : 1. **Degree $$n$$ (Even or Odd)** 2. **Sign of the Leading Coefficient $$a_n$$ (Positive or Negative)** Visible text: 1. **Degree (Even or Odd)** 2. **Sign of the Leading Coefficient (Positive or Negative)** There are four possible combinations: 1. **$$n$$ Even, $$a_n > 0$$ (Positive):** - As $$x \to \infty$$, $$y \to \infty$$ (rises right $$\nearrow$$) - As $$x \to -\infty$$, $$y \to \infty$$ (rises left $$\nwarrow$$) - Examples: $$y = x^2$$, $$y = x^4$$ Graph of $$y = x^2$$ (Even, Positive) } description="Graph rises to the left and right." showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 7 }, (_, i) => { const x = -3 + i; return { x, y: x * x, z: 0 }; }), color: getColor("LIME"), }, ]} /> 2. **$$n$$ Even, $$a_n < 0$$ (Negative):** - As $$x \to \infty$$, $$y \to -\infty$$ (falls right $$\searrow$$) - As $$x \to -\infty$$, $$y \to -\infty$$ (falls left $$\swarrow$$) - Examples: $$y = -x^2$$, $$y = -x^6$$ Graph of $$y = -x^2$$ (Even, Negative) } description="Graph falls to the left and right." showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 7 }, (_, i) => { const x = -3 + i; return { x, y: -(x * x), z: 0 }; }), color: getColor("ROSE"), }, ]} /> 3. **$$n$$ Odd, $$a_n > 0$$ (Positive):** - As $$x \to \infty$$, $$y \to \infty$$ (rises right $$\nearrow$$) - As $$x \to -\infty$$, $$y \to -\infty$$ (falls left $$\swarrow$$) - Examples: $$y = x$$, $$y = x^3$$, $$y = x^5$$ Graph of $$y = x^3$$ (Odd, Positive) } description="Graph falls to the left and rises to the right." showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 41 }, (_, i) => { const x = -3 + i * 0.15; return { x, y: x * x * x, z: 0 }; }), color: getColor("SKY"), showPoints: false, }, ]} /> 4. **$$n$$ Odd, $$a_n < 0$$ (Negative):** - As $$x \to \infty$$, $$y \to -\infty$$ (falls right $$\searrow$$) - As $$x \to -\infty$$, $$y \to \infty$$ (rises left $$\nwarrow$$) - Examples: $$y = -x$$, $$y = -x^3$$ Graph of $$y = -x^3$$ (Odd, Negative) } description="Graph rises to the left and falls to the right." showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 41 }, (_, i) => { const x = -3 + i * 0.15; return { x, y: -(x * x * x), z: 0 }; }), color: getColor("AMBER"), showPoints: false, }, ]} /> Visible text: 1. ** Even, (Positive):** - As , (rises right ) - As , (rises left ) - Examples: , Graph of (Even, Positive) } description="Graph rises to the left and right." showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 7 }, (_, i) => { const x = -3 + i; return { x, y: x * x, z: 0 }; }), color: getColor("LIME"), }, ]} /> 2. ** Even, (Negative):** - As , (falls right ) - As , (falls left ) - Examples: , Graph of (Even, Negative) } description="Graph falls to the left and right." showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 7 }, (_, i) => { const x = -3 + i; return { x, y: -(x * x), z: 0 }; }), color: getColor("ROSE"), }, ]} /> 3. ** Odd, (Positive):** - As , (rises right ) - As , (falls left ) - Examples: , , Graph of (Odd, Positive) } description="Graph falls to the left and rises to the right." showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 41 }, (_, i) => { const x = -3 + i * 0.15; return { x, y: x * x * x, z: 0 }; }), color: getColor("SKY"), showPoints: false, }, ]} /> 4. ** Odd, (Negative):** - As , (falls right ) - As , (rises left ) - Examples: , Graph of (Odd, Negative) } description="Graph rises to the left and falls to the right." showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 41 }, (_, i) => { const x = -3 + i * 0.15; return { x, y: -(x * x * x), z: 0 }; }), color: getColor("AMBER"), showPoints: false, }, ]} /> ### Using End Behavior Knowing the end behavior is very helpful for identifying the graph of a polynomial function without having to plot it in detail. **Application Example:** Match the following functions with their likely end behavior: 1. $$f(x) = x^4 + 2x^3 - 2x - 3$$ - Leading term: $$x^4$$ - Degree $$n=4$$ (Even) - Leading coefficient $$a_n=1$$ (Positive) - End behavior: Rises left ($$\nwarrow$$), Rises right ($$\nearrow$$) Graph of $$f(x) = x^4 + 2x^3 - 2x - 3$$ } description={ <> End Behavior: $$\nwarrow$${" "} $$\nearrow$$ } showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 51 }, (_, i) => { const x = -2.5 + i * (4.3 / 50); const y = x ** 4 + 2 * x ** 3 - 2 * x - 3; return { x, y, z: 0 }; }), color: getColor("TEAL"), showPoints: false, }, ]} /> 2. $$g(x) = -x^3 + 2x^2 - x + 1$$ - Leading term: $$-x^3$$ - Degree $$n=3$$ (Odd) - Leading coefficient $$a_n=-1$$ (Negative) - End behavior: Rises left ($$\nwarrow$$), Falls right ($$\searrow$$) Graph of $$g(x) = -x^3 + 2x^2 - x + 1$$ } description={ <> End Behavior: $$\nwarrow$${" "} $$\searrow$$ } showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 51 }, (_, i) => { const x = -2.5 + i * 0.1; const y = -(x ** 3) + 2 * x ** 2 - x + 1; return { x, y, z: 0 }; }), color: getColor("ORANGE"), showPoints: false, }, ]} /> 3. $$h(x) = -x^6 - \frac{11}{4}x^5 + x^4 + 5x^3 + 2$$ - Leading term: $$-x^6$$ - Degree $$n=6$$ (Even) - Leading coefficient $$a_n=-1$$ (Negative) - End behavior: Falls left ($$\swarrow$$), Falls right ($$\searrow$$) Graph of{" "} $$h(x) = -x^6 - \frac{11}{4}x^5 + x^4 + 5x^3 + 2$$ } description={ <> End Behavior: $$\swarrow$${" "} $$\searrow$$ } showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 111 }, (_, i) => { const x = -2.5 + i * (4.1 / 110); const y = -(x ** 6) - (11 / 4) * x ** 5 + x ** 4 + 5 * x ** 3 + 2; return { x, y, z: 0 }; }), color: getColor("FUCHSIA"), showPoints: false, }, ]} /> 4. $$k(x) = 25x^5 - 20x^4 - 26x^3 + 12x^2 + 9x - 1$$ - Leading term: $$25x^5$$ - Degree $$n=5$$ (Odd) - Leading coefficient $$a_n=25$$ (Positive) - End behavior: Falls left ($$\swarrow$$), Rises right ($$\nearrow$$) Graph of{" "} $$k(x) = 25x^5 - 20x^4 - 26x^3 + 12x^2 + 9x - 1$$ } description={ <> End Behavior: $$\swarrow$${" "} $$\nearrow$$ } showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 71 }, (_, i) => { const x = -1.2 + i * 0.035; const y = 25 * x ** 5 - 20 * x ** 4 - 26 * x ** 3 + 12 * x ** 2 + 9 * x - 1; return { x, y, z: 0 }; }), color: getColor("INDIGO"), showPoints: false, }, ]} /> Visible text: 1. - Leading term: - Degree (Even) - Leading coefficient (Positive) - End behavior: Rises left (), Rises right () Graph of } description={ <> End Behavior: {" "} } showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 51 }, (_, i) => { const x = -2.5 + i * (4.3 / 50); const y = x ** 4 + 2 * x ** 3 - 2 * x - 3; return { x, y, z: 0 }; }), color: getColor("TEAL"), showPoints: false, }, ]} /> 2. - Leading term: - Degree (Odd) - Leading coefficient (Negative) - End behavior: Rises left (), Falls right () Graph of } description={ <> End Behavior: {" "} } showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 51 }, (_, i) => { const x = -2.5 + i * 0.1; const y = -(x ** 3) + 2 * x ** 2 - x + 1; return { x, y, z: 0 }; }), color: getColor("ORANGE"), showPoints: false, }, ]} /> 3. - Leading term: - Degree (Even) - Leading coefficient (Negative) - End behavior: Falls left (), Falls right () Graph of{" "} } description={ <> End Behavior: {" "} } showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 111 }, (_, i) => { const x = -2.5 + i * (4.1 / 110); const y = -(x ** 6) - (11 / 4) * x ** 5 + x ** 4 + 5 * x ** 3 + 2; return { x, y, z: 0 }; }), color: getColor("FUCHSIA"), showPoints: false, }, ]} /> 4. - Leading term: - Degree (Odd) - Leading coefficient (Positive) - End behavior: Falls left (), Rises right () Graph of{" "} } description={ <> End Behavior: {" "} } showZAxis={false} cameraPosition={[0, 0, 15]} data={[ { points: Array.from({ length: 71 }, (_, i) => { const x = -1.2 + i * 0.035; const y = 25 * x ** 5 - 20 * x ** 4 - 26 * x ** 3 + 12 * x ** 2 + 9 * x - 1; return { x, y, z: 0 }; }), color: getColor("INDIGO"), showPoints: false, }, ]} /> By analyzing the leading term, we can predict the general shape of the graph at its ends.