Math Problem Statement

  1. 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) y=x2+1y = x^2 + 1

  • Base Graph: The base graph here is y=x2y = x^2, a parabola with its vertex at (0,0)(0, 0).
  • Transformation: The "+1" indicates a vertical shift upwards by 1 unit.
  • Resulting Graph: The vertex of the parabola shifts from (0,0)(0, 0) to (0,1)(0, 1).

(b) y=(x+3)2y = (x + 3)^2

  • Base Graph: The base graph is again y=x2y = x^2.
  • Transformation: The term x+3x + 3 inside the function indicates a horizontal shift to the left by 3 units.
  • Resulting Graph: The vertex of the parabola shifts from (0,0)(0, 0) to (3,0)(-3, 0).

(c) y=3(x+1)2y = 3 - (x + 1)^2

  • Base Graph: The base graph is y=x2y = x^2.
  • Transformation: The expression (x+1)(x + 1) 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 (0,0)(0, 0) to (1,3)(-1, 3), and the parabola opens downwards.

(d) y=2(x+2)2y = 2 - (x + 2)^{-2}

  • Base Graph: The base graph is y=x2y = x^{-2} (which is a hyperbola).
  • Transformation: The term (x+2)(x + 2) shifts the graph left by 2 units. The coefficient 22 indicates a vertical shift upwards by 2 units.
  • Resulting Graph: The graph of y=x2y = x^{-2} moves left by 2 units and up by 2 units.

Summary of Translations:

  1. (a) y=x2+1y = x^2 + 1: Shift up by 1 unit. Vertex at (0,1)(0, 1).
  2. (b) y=(x+3)2y = (x + 3)^2: Shift left by 3 units. Vertex at (3,0)(-3, 0).
  3. (c) y=3(x+1)2y = 3 - (x + 1)^2: Shift left by 1 unit, reflect over the x-axis, and shift up by 3 units. Vertex at (1,3)(-1, 3).
  4. (d) y=2(x+2)2y = 2 - (x + 2)^{-2}: 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:

  1. How would the graph of y=x2y = x^2 change if it were shifted right instead of left?
  2. What effect does multiplying the function by a negative constant have on the graph of a quadratic function?
  3. How can you determine the direction in which a parabola opens based on its equation?
  4. What is the general effect of a shift inside a function versus a shift outside the function?
  5. How can you graph y=(x1)2+2y = (x - 1)^2 + 2 using transformations?

Tip: Always identify whether transformations are inside or outside the function. Inside (affecting xx) leads to horizontal shifts, while outside (affecting yy) 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