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What is the Significance of 'k' in y = a(x-h)² + k?
Grade Level:
Class 9
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
In the equation y = a(x-h)² + k, 'k' represents the vertical shift of the parabola from its standard position. It tells us how much the graph moves up or down along the y-axis, directly determining the y-coordinate of the vertex.
Simple Example
Quick Example
Imagine a cricket ball thrown upwards. Its path forms a parabola. If 'k' is positive, the ball goes higher. If 'k' is negative, it goes lower. So, 'k' basically tells us the maximum or minimum height the ball reaches.
Worked Example
Step-by-Step
Let's find the vertex of the parabola y = 2(x-3)² + 5.
Step 1: Identify the standard form of the parabola equation: y = a(x-h)² + k.
---Step 2: Compare our given equation y = 2(x-3)² + 5 with the standard form.
---Step 3: From the comparison, we can see that a = 2, h = 3, and k = 5.
---Step 4: The vertex of a parabola in this form is (h, k).
---Step 5: Substitute the values of h and k into the vertex formula.
---Step 6: So, the vertex is (3, 5).
Answer: The vertex of the parabola y = 2(x-3)² + 5 is (3, 5). The value of 'k' (which is 5) tells us the y-coordinate of this vertex.
Why It Matters
Understanding 'k' helps engineers design bridges and roller coasters by controlling their highest or lowest points. In data science, it helps analyze trends and predict peaks or valleys in data, like predicting stock market highs or lows. Even in AI, 'k' can help optimize functions to find best solutions.
Common Mistakes
MISTAKE: Students confuse 'k' with 'h' and think 'k' controls horizontal movement. | CORRECTION: 'k' controls vertical movement (up/down), while 'h' controls horizontal movement (left/right).
MISTAKE: Forgetting that 'k' directly gives the y-coordinate of the vertex. | CORRECTION: The vertex of the parabola y = a(x-h)² + k is always (h, k). 'k' is the y-coordinate.
MISTAKE: Changing the sign of 'k' when identifying it from the equation. | CORRECTION: The sign of 'k' is taken directly as it appears in the equation. If it's '+k', it's positive; if it's '-k', it's negative. For example, in y = 2(x-1)² - 4, k = -4.
Practice Questions
Try It Yourself
QUESTION: What is the y-coordinate of the vertex for the parabola y = 5(x+2)² + 7? | ANSWER: 7
QUESTION: A parabola has the equation y = -3(x-1)² - 9. What is the value of 'k' and what does it tell us about the parabola's vertex? | ANSWER: k = -9. It tells us that the y-coordinate of the vertex is -9, meaning the parabola's highest point is at y = -9 (since 'a' is negative).
QUESTION: If a parabola's vertex is at (4, -2) and its equation is of the form y = a(x-h)² + k, what are the values of 'h' and 'k'? | ANSWER: h = 4, k = -2
MCQ
Quick Quiz
In the equation y = a(x-h)² + k, what does 'k' represent?
Horizontal shift
Vertical shift
Stretch or compression factor
Direction of opening
The Correct Answer Is:
B
k' directly adds or subtracts a value from 'y', causing the entire graph to move up or down on the y-axis, which is a vertical shift. 'h' is for horizontal shift, 'a' for stretch/compression and direction.
Real World Connection
In the Real World
Think about the path of a javelin thrown by an athlete in a sports meet. The highest point the javelin reaches can be determined using the 'k' value in the parabolic equation that describes its flight. This helps coaches analyze performance and improve throwing techniques.
Key Vocabulary
Key Terms
PARABOLA: A U-shaped curve that is the graph of a quadratic equation | VERTEX: The highest or lowest point on a parabola | VERTICAL SHIFT: Moving a graph up or down along the y-axis | Y-AXIS: The vertical line in a coordinate plane
What's Next
What to Learn Next
Now that you understand 'k', you should explore the significance of 'h' in y = a(x-h)² + k. Understanding 'h' will complete your knowledge of how the vertex form of a parabola works and how to graph it effectively. Keep exploring!
