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Questions and Answers
If a function $f(x) = 3x^2 - 2x + 5$ is transformed by applying a vertical shift of 4 units upward, what is the new function?
If a function $f(x) = 3x^2 - 2x + 5$ is transformed by applying a vertical shift of 4 units upward, what is the new function?
- $f(x) = 3x^2 - 2x + 9$ (correct)
- $f(x) = 3x^2 - 2x + 1$
- $f(x) = 3x^2 - 2x + 8$
- $f(x) = 3x^2 - 2x + 7$
A rectangle has a length that is 3 times its width. If the perimeter of the rectangle is 48 units, what is the width?
A rectangle has a length that is 3 times its width. If the perimeter of the rectangle is 48 units, what is the width?
- 4 units
- 8 units (correct)
- 12 units
- 6 units
What is the area of a triangle with a base of 10 units and a height of 5 units?
What is the area of a triangle with a base of 10 units and a height of 5 units?
- 15 square units
- 50 square units
- 30 square units
- 25 square units (correct)
A student scored 75%, 85%, and 90% on three tests. What score does the student need on a fourth test to achieve an average of 80%?
A student scored 75%, 85%, and 90% on three tests. What score does the student need on a fourth test to achieve an average of 80%?
If the equation of a line is given by $y = 2x + 3$, what is the slope of the line?
If the equation of a line is given by $y = 2x + 3$, what is the slope of the line?
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Study Notes
Function Transformation
- Original function: ( f(x) = 3x^2 - 2x + 5 )
- Vertical shift of 4 units upward alters the function to: ( f(x) + 4 = 3x^2 - 2x + 9 )
Rectangle Perimeter and Dimensions
- Rectangle's length is 3 times its width: Let width be ( w ), then length = ( 3w )
- Perimeter formula: ( P = 2(\text{length} + \text{width}) ) gives ( 48 = 2(3w + w) )
- Simplified to ( 48 = 8w ), yielding ( w = 6 ) units for the width
Area of Triangle
- Area formula: ( A = \frac{1}{2} \times \text{base} \times \text{height} )
- For base of 10 units and height of 5 units: ( A = \frac{1}{2} \times 10 \times 5 = 25 ) square units
Average Score Calculation
- Scores: 75%, 85%, 90% on three tests
- Total score from three tests: ( 75 + 85 + 90 = 250 )
- To achieve an average of 80% across four tests: ( \text{Target total score} = 4 \times 80 = 320 )
- Required score on fourth test: ( 320 - 250 = 70 )
Slope of a Line
- Given line equation: ( y = 2x + 3 )
- The slope-intercept form indicates the slope is ( 2 )
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