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Questions and Answers
Given $f(x) = x^3 - 6x^2 + 5$, after a translation 1 unit left, a reflection in the x-axis, and a vertical stretch by a factor of 2, the resulting rule for $g(x)$ will have a leading term of ______.
Given $f(x) = x^3 - 6x^2 + 5$, after a translation 1 unit left, a reflection in the x-axis, and a vertical stretch by a factor of 2, the resulting rule for $g(x)$ will have a leading term of ______.
-2x^3
Given $f(x) = 3x^4 + x^3 + 3x^2 + 12$, after a horizontal shrink by a factor of $\frac{1}{3}$, a translation 8 units down, followed by a reflection in the y-axis, the constant term of the equation representing $g(x)$ is ______.
Given $f(x) = 3x^4 + x^3 + 3x^2 + 12$, after a horizontal shrink by a factor of $\frac{1}{3}$, a translation 8 units down, followed by a reflection in the y-axis, the constant term of the equation representing $g(x)$ is ______.
4
Given $f(x) = x^3 - 6$, after translating 3 units left and reflecting in the y-axis, the coefficient of the $x^2$ term in $g(x)$ is ______.
Given $f(x) = x^3 - 6$, after translating 3 units left and reflecting in the y-axis, the coefficient of the $x^2$ term in $g(x)$ is ______.
9
Given $f(x) = x^2 + 2x + 6$, after a vertical stretch by a factor of 2, followed by a translation of 4 units right, the coefficient of the x term in $g(x)$ is ______.
Given $f(x) = x^2 + 2x + 6$, after a vertical stretch by a factor of 2, followed by a translation of 4 units right, the coefficient of the x term in $g(x)$ is ______.
Given $f(x) = x^3 + 2x^2 - 9$, after a horizontal stretch of 2 and a translation of 2 units up, followed by a reflection in the x-axis, the coefficient of the $x^3$ term in $g(x)$ is ______.
Given $f(x) = x^3 + 2x^2 - 9$, after a horizontal stretch of 2 and a translation of 2 units up, followed by a reflection in the x-axis, the coefficient of the $x^3$ term in $g(x)$ is ______.
Given $f(x) = 2x^2 + 6x - 9$, after translating 1 unit left, reflecting over the y-axis, with a translation down 3, the coefficient of the $x^2$ term in $g(x)$ is ______.
Given $f(x) = 2x^2 + 6x - 9$, after translating 1 unit left, reflecting over the y-axis, with a translation down 3, the coefficient of the $x^2$ term in $g(x)$ is ______.
A transformation involving a vertical stretch by a factor of 2 changes the graph of $f(x) = x^2 + 2x + 6$ to $g(x)$. The constant term of $g(x)$ is ______ times the constant term of $f(x)$.
A transformation involving a vertical stretch by a factor of 2 changes the graph of $f(x) = x^2 + 2x + 6$ to $g(x)$. The constant term of $g(x)$ is ______ times the constant term of $f(x)$.
Given $f(x) = x^3 - 6$, performing a translation of 3 units left and then reflecting in the y-axis results in a function $g(x)$. The new constant term of $g(x)$ is ______.
Given $f(x) = x^3 - 6$, performing a translation of 3 units left and then reflecting in the y-axis results in a function $g(x)$. The new constant term of $g(x)$ is ______.
When $f(x) = 3x^4 + x^3 + 3x^2 + 12$ undergoes a reflection in the y-axis, the terms with ______ exponents will change their sign in the new function.
When $f(x) = 3x^4 + x^3 + 3x^2 + 12$ undergoes a reflection in the y-axis, the terms with ______ exponents will change their sign in the new function.
For $f(x) = x^2 + 2x + 6$, a horizontal shift of 4 units to the right affects the location of its vertex. Therefore, the x-coordinate changes by ______ units.
For $f(x) = x^2 + 2x + 6$, a horizontal shift of 4 units to the right affects the location of its vertex. Therefore, the x-coordinate changes by ______ units.
Considering the function $f(x) = x^3 + 2x^2 - 9$, reflecting it in the x-axis will change whether the $x^3$ term is ______.
Considering the function $f(x) = x^3 + 2x^2 - 9$, reflecting it in the x-axis will change whether the $x^3$ term is ______.
When $f(x) = 2x^2 + 6x - 9$ is translated 1 unit left and also translated down 3, only the ______ is affected.
When $f(x) = 2x^2 + 6x - 9$ is translated 1 unit left and also translated down 3, only the ______ is affected.
If $f(x) = x^3 - 6x^2 + 5$ is reflected about the x-axis, the sign of the $x^3$ term will ______.
If $f(x) = x^3 - 6x^2 + 5$ is reflected about the x-axis, the sign of the $x^3$ term will ______.
Given $f(x) = 3x^4 + x^3 + 3x^2 + 12$, shrinking the function horizontally towards the y-axis increases ______ of the graph.
Given $f(x) = 3x^4 + x^3 + 3x^2 + 12$, shrinking the function horizontally towards the y-axis increases ______ of the graph.
If $f(x) = x^3 - 6$ is first translated horizontally and then reflected across the y-axis, the ______ of the transformation matters for the $x^3$ term.
If $f(x) = x^3 - 6$ is first translated horizontally and then reflected across the y-axis, the ______ of the transformation matters for the $x^3$ term.
If $f(x) = x^2 + 2x + 6$ is vertically stretched and shifted horizontally, the ______ of the parabola will change.
If $f(x) = x^2 + 2x + 6$ is vertically stretched and shifted horizontally, the ______ of the parabola will change.
Given $f(x) = x^3 + 2x^2 - 9$, stretching the function horizontally affects how quickly it increases in value, changing its vertical ______.
Given $f(x) = x^3 + 2x^2 - 9$, stretching the function horizontally affects how quickly it increases in value, changing its vertical ______.
For $f(x) = 2x^2 + 6x - 9$, translating left stretches the curve. Translating down also stretches the curve. Therefore, the ______ of the vertex will shift.
For $f(x) = 2x^2 + 6x - 9$, translating left stretches the curve. Translating down also stretches the curve. Therefore, the ______ of the vertex will shift.
Combining a vertical stretch and a ______ transformation alters the shape of a graph by changing both steepness and horizontal positioning.
Combining a vertical stretch and a ______ transformation alters the shape of a graph by changing both steepness and horizontal positioning.
A reflection across the x-axis changes the sign and ______, while a translation shifts its position. They both modify the graph.
A reflection across the x-axis changes the sign and ______, while a translation shifts its position. They both modify the graph.
Flashcards
Translation (left/right)
Translation (left/right)
Shifts the graph horizontally
Reflection (x or y axis)
Reflection (x or y axis)
Flips the graph over a line
Vertical Stretch/Compression
Vertical Stretch/Compression
Stretches or compresses vertically
Reflection in the y-axis
Reflection in the y-axis
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Vertical Translation
Vertical Translation
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Horizontal Stretch
Horizontal Stretch
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Study Notes
- These notes cover function transformations, focusing on how to write rules for transformed graphs based on the original function, f(x).
- The transformations include translations, reflections, and stretches/shrinks.
Problem 1: f(x) = x³ - 6x² + 5
- Original function: f(x) = x³ - 6x² + 5.
- Transformation 1: Translation 1 unit left → replace x with (x + 1).
- Transformation 2: Reflection in the x-axis → multiply the function by -1.
- Transformation 3: Vertical stretch by a factor of 2 → multiply the function by 2.
- Resulting g(x): g(x) = 2[-(x+1)³ + 6(x+1)² - 5] which simplifies to g(x) = -2x³ + 6x² + 18x.
Problem 2: f(x) = 3x⁴ + x³ + 3x² + 12
- Original function: f(x) = 3x⁴ + x³ + 3x² + 12.
- Transformation 1: Horizontal shrink by a factor of 1/3 → replace x with 3x.
- Transformation 2: Translation 8 units down → subtract 8 from the function.
- Transformation 3: Reflection in the y-axis → replace x with -x.
- Resulting g(x): g(x) = 3(-x)⁴ + (-x)³ + 3(-x)² + 12 - 8 and simplifies to g(x) = 243x⁴ - 27x³ + 27x² + 4.
Problem 3: f(x) = x³ - 6
- Original function: f(x) = x³ - 6.
- Transformation 1: Translation 3 units left → replace x with (x+3).
- Transformation 2: Reflection in the y-axis → replace x with -x.
- Resulting g(x): g(x) = (-x)³ + 9(-x)² - 27(-x) + 21, which simplifies to g(x) = -x³ + 9x² - 27x + 21.
Problem 4: f(x) = x² + 2x + 6
- Original function: f(x) = x² + 2x + 6.
- Transformation 1: Vertical stretch by a factor of 2 → multiply the function by 2.
- Transformation 2: Translation 4 units right → replace x with (x - 4).
- Resulting g(x): g(x) = 2((x - 4)² + 2(x - 4) + 6) = 2x² -12x + 28.
Problem 5: f(x) = x³ + 2x² - 9
- Original function: f(x) = x³ + 2x² - 9.
- Transformation 1: Horizontal stretch of 2 → replace x with (1/2)x.
- Transformation 2: Translation 2 units up → add 2 to the function.
- Transformation 3: Reflection in the x-axis → multiply the function by -1.
- Resulting g(x): g(x) = -((1/2x)³ + 2((1/2)x²) - 9 + 2) = -1/8x³ - 1/2x² + 7.
Problem 6: f(x) = 2x² + 6x - 9
- Original function: f(x) = 2x² + 6x - 9.
- Transformation 1: Translation 1 unit left → replace x with (x+1).
- Transformation 2: Reflection in the y-axis → replace x with -x.
- Transformation 3: Translation 3 units down → subtract 3 from the function.
- Resulting g(x): g(x) = 2(-x)² + 10(-x) - 1 - 3 = 2x² - 10x - 4.
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