Math Problem Statement
- Use the rules of shifting of graphs, and sketch the following graphs. (a) y = x 2 + 1 (b) y = (x + 3)2 (c) y = 3 − (x + 1)2 (d) y = 2 − (x + 2)−2
Solution
Let's go through each of the given functions one by one, using the rules of shifting graphs (translation, reflection, scaling).
(a)
- Base Graph: The base graph here is , a parabola with its vertex at .
- Transformation: The "+1" indicates a vertical shift upwards by 1 unit.
- Resulting Graph: The vertex of the parabola shifts from to .
(b)
- Base Graph: The base graph is again .
- Transformation: The term inside the function indicates a horizontal shift to the left by 3 units.
- Resulting Graph: The vertex of the parabola shifts from to .
(c)
- Base Graph: The base graph is .
- Transformation: The expression indicates a shift to the left by 1 unit, and the minus sign in front of the squared term indicates a reflection over the x-axis. Finally, the "+3" means a vertical shift upwards by 3 units.
- Resulting Graph: The vertex moves from to , and the parabola opens downwards.
(d)
- Base Graph: The base graph is (which is a hyperbola).
- Transformation: The term shifts the graph left by 2 units. The coefficient indicates a vertical shift upwards by 2 units.
- Resulting Graph: The graph of moves left by 2 units and up by 2 units.
Summary of Translations:
- (a) : Shift up by 1 unit. Vertex at .
- (b) : Shift left by 3 units. Vertex at .
- (c) : Shift left by 1 unit, reflect over the x-axis, and shift up by 3 units. Vertex at .
- (d) : Shift left by 2 units and up by 2 units.
Would you like to dive deeper into any of these transformations? Or is there a particular step you'd like more detail on?
Related questions:
- How would the graph of change if it were shifted right instead of left?
- What effect does multiplying the function by a negative constant have on the graph of a quadratic function?
- How can you determine the direction in which a parabola opens based on its equation?
- What is the general effect of a shift inside a function versus a shift outside the function?
- How can you graph using transformations?
Tip: Always identify whether transformations are inside or outside the function. Inside (affecting ) leads to horizontal shifts, while outside (affecting ) leads to vertical shifts.
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Math Problem Analysis
Mathematical Concepts
Graph Transformations
Quadratic Functions
Hyperbolic Functions
Formulas
y = x^2 + 1
y = (x + 3)^2
y = 3 - (x + 1)^2
y = 2 - (x + 2)^{-2}
Theorems
Translation of graphs
Reflection over the x-axis
Vertical and horizontal shifts
Suitable Grade Level
Grades 9-11
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