Graphs and Transformation Cheat Sheet - Cubic & Quartic Graphs - Sketching, Intersections, and Transformations

Q: How do you determine the general shape of a cubic graph from its equation?

Compare the cubic function to a general form and use the parameter conditions to infer the graph’s overall shape. The guide specifies different shape behavior depending on whether the leading coefficient (e.g., a) is positive or negative.

Q: How are x-intercepts found when sketching a cubic function?

Set the function equal to 0 to solve for the x-values, then substitute back to confirm intercept coordinates. The document’s cubic example factors the equation to obtain roots and lists the x-intercepts.

Q: What transformations can be applied to translate, stretch, or reflect graphs?

Translation uses a constant added outside (vertical shift) or inside (horizontal shift) the function. Stretching scales the graph vertically or horizontally, while multiplying by -1 or replacing x with -x produces reflections across the x-axis or y-axis respectively.

Oct 13, 2026

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Concise math cheat sheet covering cubic and quartic function graphing, including their standard algebraic forms and how coefficients determine key shapes. Explains root-finding for intercepts, illustrating cubic examples by solving a related quadratic factorization and locating x-intercepts. Extends to quartic sketching, reciprocal graphs via asymptotes, and determining intersection points by equating functions and solving. Also details graph translation (vertical/horizontal shifts), stretching and reflection rules, and transforming unfamiliar functions using intercepts, turning points, and intersection behavior.

LanguageEnglish

CategoryPresentations

FormatPDF

Pages1

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Cubic & Quartic Graphs

Sketching Cubic Functions

Root-Finding and Intercepts

Sketching Quartic and Reciprocal Graphs

Points of Intersection

Translating Graphs

Stretching and Reflection

Transforming Graphs with Key Points

Graphs and Transformation Cheat Sheet Cubic & Quartic Graphs Cubic functions come in the form of 􀝂 (􀝔) = 􀜽􀝔3 + 􀜾􀝔2 + 􀜿􀝔 + 􀝀 . Quartic functions come in the form of 􀝂 (􀝔) = 􀜽􀝔4 + 􀜾􀝔3 + 􀜿􀝔2 + 􀝀􀝔 + 􀝁 , where 􀜽, 􀜾 . 􀜿, 􀝀 & 􀝁 are all real numbers and where 􀜽 ≠ 0. Similar to sketching quadratic graphs, cubic graphs can also be represented on a plot. There are two main basic shapes for cubic graphs if 􀜽 > 0: 􀜽 < 0: Example 1: Sketch the curve with the equation 􀝕 = 􀝔 3 + 􀝔 2 − 2􀝔 . Before we start sketching, we need to know • the general shape of the cubic equation • the location of the roots of the equation We can compare the function to a general function of 􀝂(􀝔) = 􀜽􀝔3 + 􀜾􀝔2 + 􀜿􀝔 + 􀝀 to determine the shape of the curve. 􀜽 = 1 and 1 > 0 hence the shape of the graph would look like . The next step would be to find the roots of the functions so we can determine where it would cross the 􀝔 axis. To do this we need to solve the quadratic equation of 􀝔 3 + 􀝔 2 − 2􀝔 = 0 􀝔 (􀝔2 + 􀝔 − 2) = 􀝔 (􀝔 + 2)(􀝔 − 1) = 0 􀝔 = 0 􀜽􀝊􀝀 􀝔 − 1 = 0 􀜽􀝊􀝀 􀝔 + 2 = 0 Hence the roots of the function are at: 􀝔 = 0 􀜽􀝊􀝀 􀝔 = 1 􀜽􀝊􀝀 􀝔 = −2 At coordinates (0,0), (1,0) & (−2,0) are where the x intercepts will be located. The same method can be applied to sketching graphs of quartic functions. The basic shapes of these graphs come in the form of: . Reciprocal graphs Reciprocal graphs that come in the form of 􀝂(􀝔) = 􀯔􀯫 or 􀝂(􀝔) = 􀯔􀯫2, where 􀜽 is any real number, can also be sketched by considering their asymptotes. The graphs of 􀝕 = 􀯔􀯫 and 􀝕 = 􀯔􀯫2 both have asymptotes at 􀝔 = 0 and 􀝕 = 0. The basic shapes of these reciprocal graphs can be illustrated as: 􀝕 = 􀜽 􀝓ℎ􀝁􀝎􀝁 􀜽 > 0 􀝔 , | 􀝕 = − 􀜽
~~ ~~ 􀝓ℎ􀝁􀝎􀝁 􀜽 < 0 􀝔 ,
| 􀝕 = 􀜽 􀝓ℎ􀝁􀝎􀝁 􀜽 > 0 􀝔 2 ,
| | --- | --- | 􀝕 􀜽 = − 􀝔 2 , 􀝓ℎ􀝁􀝎􀝁 􀜽 < 0 Points of Intersection Multiple functions can be sketched on a single graph to show the points of intersection, which represent the solutions to respective equations. Example 2: Sketch the following functions and find the 􀝔 coordinate of the intersection points. 􀝂 (􀝔) = 3􀝔 − 􀝔 2 􀝃 (􀝔) = 2􀝔3 − 􀝔 2 The curves have been sketched using the methods you have learnt in this course. To find the intersection point, we must obtain the 􀝔 coordinate of the locations. To solve this, we need to find the solutions to • 􀝂 (􀝔) = 􀝃 (􀝔) 3􀝔 − 􀝔 2 = 2􀝔3 − 􀝔 2 2􀝔3 − 3􀝔 = 0 􀝔 (2􀝔2 − 3) = 0 􀝔 = 0 & 2􀝔2 − 3 = 0 􀝔 = 0, 􀝔 = √ ~~ ~~32 & 􀝔 = −√ ~~ ~~32 Translating graphs Adding or subtracting a constant outside, 􀝕 = 􀝂 (􀝔) + 􀜽, or inside, 􀝕 = 􀝂 (􀝔 + 􀜽), a function can translate a graph vertically or horizontally respectively. Note that when translating functions, the asymptote of that function is also translated if it has one. • The translation of 􀝕 = 􀝂 (􀝔) + 􀜽 can be represented by the vector (0􀜽)−􀜽 • The translation of 􀝕 = 􀝂 (􀝔 + 􀜽) can be represented by the vector ( 0 ) Example 3: Given that 􀝂(􀝔) = 􀝔 2 , sketch the curve of 􀝕 = 􀝂 (􀝔 − 3) . This is the graph of 􀝕 = 􀝔 2 Applying the translation of 􀝕 = 􀝂 (􀝔 − 3) would shift the graph by three units to the right, forming anew graph of 􀝕 = (􀝔 − 3)2 Pure Year 1 Stretching Graphs Similar to translating graphs, multiplying a constant outside, 􀝕 = 􀜽􀝂 (􀝔), or inside, 􀝕 =􀝂 (􀜽􀝔), a function stretches the graph in the vertical direction or horizontal direction respectively. • 􀝕 = 􀜽􀝂(􀝔) would stretch the graph in the vertical direction by a multiple of 􀜽 . • 􀝕 = 􀝂 (􀜽􀝔) would stretch the graph in the horizontal direction by a multiple of 1 􀯔 . • 􀝕 = −􀝂(􀝔) would be the reflection of 􀝕 = 􀝂 (􀝔) in the 􀝔-axis. • 􀝕 = 􀝂 (−􀝔) would be the reflection of 􀝕 = 􀝂 (􀝔) in the 􀝕-axis. Example 4: Given that 􀝂(􀝔) = 16 − 4􀝔2, sketch the curve with the equation 􀝕 = 12 􀝂 (􀝔) . 1 Before we start to sketch 􀝕 = 􀝂 (􀝔), we need to know the 􀝔 and 􀝕 intercepts of the 2 curve 􀝕 = 􀝂 (􀝔) and how the curve looks like. 􀝕 = 16 − 4􀝔2 􀝕 = (4 − 2􀝔)(4 + 2􀝔) To find the 􀝔 intercept we need to equate 􀝕 as 0. 0

FAQs

How do you determine the general shape of a cubic graph from its equation?

Compare the cubic function to a general form and use the parameter conditions to infer the graph’s overall shape. The guide specifies different shape behavior depending on whether the leading coefficient (e.g., a) is positive or negative.

How are x-intercepts found when sketching a cubic function?

Set the function equal to 0 to solve for the x-values, then substitute back to confirm intercept coordinates. The document’s cubic example factors the equation to obtain roots and lists the x-intercepts.

What transformations can be applied to translate, stretch, or reflect graphs?

Translation uses a constant added outside (vertical shift) or inside (horizontal shift) the function. Stretching scales the graph vertically or horizontally, while multiplying by -1 or replacing x with -x produces reflections across the x-axis or y-axis respectively.

Oct 13, 2026

Upload by Callum