Math Problem Statement
Graph the following: y = (x - 3)^2, y = (x + 2)^2 + 4
Solution
The two equations you provided are quadratic functions, and both represent parabolas. Let's analyze and describe how to graph each one.
1. Equation 1:
- This is a standard form of a parabola, , where:
- (shifts the graph horizontally to the right by 3 units),
- (no vertical shift).
- The vertex of the parabola is at , and since the coefficient of is positive, the parabola opens upwards.
- The axis of symmetry is .
- This graph is symmetrical around , and its shape is the same as the standard parabola , just shifted.
2. Equation 2:
- This is also in the form , where:
- (shifts the graph horizontally to the left by 2 units),
- (shifts the graph vertically up by 4 units).
- The vertex of this parabola is at , and it also opens upwards because the coefficient of is positive.
- The axis of symmetry is .
Both graphs represent parabolas that open upwards, but their vertices are different due to horizontal and vertical shifts.
Would you like a graphical representation or more details on how to draw the graph on the grid?
5 Questions for Expansion:
- How do we determine the direction of a parabola (up or down)?
- What is the effect of changing the sign of the square term, for example ?
- How can we find the points where these parabolas intersect the x-axis or y-axis?
- What are some applications of quadratic functions like these in real-world problems?
- How would the graph change if we added a coefficient in front of the squared term, like ?
Tip:
The vertex form of a quadratic function is useful for easily identifying the vertex and direction of the parabola, making graphing simpler.
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Graphing
Parabolas
Formulas
Vertex form of quadratic equation: y = (x - h)^2 + k
Theorems
Vertex theorem for parabolas
Suitable Grade Level
Grades 9-12