Function transformations
51 flashcards covering Function transformations for the SAT Math section.
Function transformations involve taking a basic mathematical function, like a line or a parabola, and modifying it to create a new one. Imagine a graph of a simple function as a shape on a coordinate plane; transformations shift, stretch, flip, or resize that shape without changing its fundamental type. For example, adding a number to the function might move the graph up or down, while multiplying inside the parentheses could slide it left or right. These changes help us understand how equations represent real-world patterns, such as growth or reflections.
On the SAT Math section, function transformations often appear in questions about graphing or interpreting equations, typically in multiple-choice formats that ask you to identify how a graph shifts or to predict the new equation. Common traps include confusing horizontal and vertical shifts—remember, changes inside the function affect the input, while those outside affect the output—or overlooking how negative signs cause reflections. Focus on practicing with standard forms like f(x) + k or a*f(x-h), and always sketch quick graphs to visualize the effects.
A concrete tip: Practice transforming basic functions like y = x^2 to build intuition.
Terms (51)
- 01
Vertical shift
A vertical shift moves the graph of a function up or down by adding or subtracting a constant to the entire function.
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Horizontal shift
A horizontal shift moves the graph of a function left or right by replacing x with x minus or plus a constant inside the function.
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Vertical stretch
A vertical stretch enlarges the graph of a function vertically by multiplying the entire function by a constant greater than 1.
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Vertical compression
A vertical compression shrinks the graph of a function vertically by multiplying the entire function by a constant between 0 and 1.
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Horizontal stretch
A horizontal stretch widens the graph of a function by replacing x with a fraction of x, such as x divided by a constant greater than 1.
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Horizontal compression
A horizontal compression narrows the graph of a function by replacing x with a multiple of x, such as a constant greater than 1 times x.
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Reflection over x-axis
Reflecting a graph over the x-axis produces a mirror image by multiplying the entire function by -1, flipping it upside down.
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Reflection over y-axis
Reflecting a graph over the y-axis produces a mirror image by replacing x with -x inside the function, flipping it left to right.
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Parent function
A parent function is the basic, unaltered form of a function, such as f(x) = x^2, from which other functions are derived through transformations.
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Transformed function
A transformed function is the result of applying shifts, stretches, compressions, or reflections to a parent function, altering its graph.
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f(x) + k transformation
The transformation f(x) + k shifts the graph of f(x) vertically: up by k units if k is positive, or down if k is negative.
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f(x - h) transformation
The transformation f(x - h) shifts the graph of f(x) horizontally: right by h units if h is positive, or left if h is negative.
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a f(x) transformation
Multiplying f(x) by a constant a stretches or compresses the graph vertically: stretches if |a| > 1, compresses if 0 < |a| < 1, and reflects over x-axis if a is negative.
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f(bx) transformation
Replacing x with bx in f(x) stretches or compresses the graph horizontally: compresses if |b| > 1, stretches if 0 < |b| < 1, and reflects over y-axis if b is negative.
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f(x) transformation
The transformation -f(x) reflects the graph of f(x) over the x-axis, creating a mirror image below the x-axis.
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f(-x) transformation
The transformation f(-x) reflects the graph of f(x) over the y-axis, creating a mirror image to the left of the y-axis.
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Order of transformations
The order in which transformations are applied can affect the final graph, so it's important to apply them in the sequence they appear in the equation.
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Effect on vertex of quadratic
For a quadratic function, transformations like f(x - h) + k move the vertex to (h, k), altering the graph's position.
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Effect on y-intercept
Transformations such as vertical shifts or stretches change the y-intercept by altering the function's value at x = 0.
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Effect on x-intercept
Horizontal shifts or stretches can change the x-intercepts by moving or scaling the points where the graph crosses the x-axis.
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Symmetry in functions
Symmetry refers to how a function's graph behaves under reflections, such as even functions symmetric about the y-axis and odd functions symmetric about the origin.
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Even function
An even function satisfies f(-x) = f(x), meaning its graph is symmetric with respect to the y-axis, like a reflection over the y-axis.
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Odd function
An odd function satisfies f(-x) = -f(x), meaning its graph is symmetric with respect to the origin, like a 180-degree rotation.
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Common trap: Vertical vs. horizontal shift
A common error is confusing vertical shifts, which affect the output, with horizontal shifts, which affect the input, leading to incorrect graph movements.
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How transformations affect domain
Horizontal transformations like stretches or compressions can change the domain if they alter the input values, but vertical ones do not.
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How transformations affect range
Vertical transformations like stretches or shifts can change the range by altering output values, while horizontal ones typically do not.
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Transformations of quadratic functions
For quadratic functions like f(x) = x^2, transformations shift, stretch, or reflect the parabola while maintaining its U-shape.
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Transformations of linear functions
For linear functions like f(x) = x, transformations result in lines with the same slope but shifted or stretched positions.
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Transformations of absolute value functions
For absolute value functions like f(x) = |x|, transformations move the V-shaped graph while preserving its vertex and angles.
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Transformations of square root functions
For square root functions like f(x) = √x, transformations shift or stretch the graph, which starts at the origin and increases slowly.
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Graph of y = (x-2)^2 + 3
This is a quadratic function shifted right by 2 units and up by 3 units from the parent y = x^2, with vertex at (2, 3).
The graph is a parabola opening upwards, crossing the y-axis at 7.
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Strategy for sketching transformed graphs
To sketch a transformed graph, start with the parent function, apply transformations step by step, and plot key points like intercepts and vertices.
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Identifying transformations from equation
From an equation like y = 2f(x - 1) + 3, identify a horizontal shift right by 1, a vertical stretch by 2, and a vertical shift up by 3.
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Writing equation from transformed graph
Given a graph that is the parent function shifted or stretched, write the equation by determining the transformations applied to the original.
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Common trap: Reflection and stretches
Multiplying by a negative constant both stretches and reflects, so remember it combines two effects, not just one.
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Effect of multiple transformations
When multiple transformations are combined, such as in y = -2f(x + 1), apply them in order: shift, stretch, and reflect.
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Transformations and asymptotes
For functions with asymptotes, like exponentials, transformations shift the asymptotes accordingly without changing their orientation.
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Inverse function transformation
Finding the inverse of a function involves swapping x and y, which is like reflecting the graph over the line y = x.
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Graph of y = |x + 1| - 2
This absolute value function is shifted left by 1 unit and down by 2 units from y = |x|, with vertex at (-1, -2).
It crosses the x-axis at x = -3 and x = 1.
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Horizontal asymptote shift
For exponential functions, horizontal shifts move the asymptote left or right, while vertical shifts move it up or down.
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Vertical asymptote and transformations
Transformations can shift vertical asymptotes horizontally for rational functions, but SAT typically focuses on basic cases.
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Composition of functions as transformation
Composing functions, like f(g(x)), can act as a transformation, altering the input before applying the original function.
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Effect on period of periodic functions
For trigonometric functions on SAT, horizontal stretches change the period, making waves wider or narrower.
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Graph of y = √(x + 3) - 1
This square root function is shifted left by 3 units and down by 1 unit from y = √x, starting at (-3, -1).
It passes through (0, √3 - 1) approximately.
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Common trap: Scaling factors
When a scaling factor is greater than 1, it's a stretch; between 0 and 1, it's a compression, but direction matters for reflections.
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Transformations of exponential functions
For exponential functions like f(x) = 2^x, transformations shift, stretch, or reflect the curve while keeping its growth rate.
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Restricted domain after transformation
Horizontal transformations might restrict the domain further, like shifting a square root function into negative values.
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Symmetry and transformations
Transformations can preserve or alter symmetry; for example, reflecting an even function keeps it even.
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Graph of y = 3x + 2
This linear function is the parent y = x stretched vertically by 3 and shifted up by 2, resulting in a steeper line.
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Inverse of a transformed function
The inverse of a transformed function requires reversing each transformation in the original equation.
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Effect on slope of linear functions
Vertical stretches change the slope of linear functions, while horizontal stretches alter it inversely.