Vertex Form and Parabola Transformations
Vertex form writes a quadratic as y = a(x − h)² + k, where (h, k) is the vertex, so y = 2(x − 3)² − 1 has vertex (3, −1). In parabola transformations, h moves the graph left or right and k up or down.
Video Lesson
Watch and learn the basics
🎬 Did this video explain it clearly?
Flashcards
Review key concepts visually
%20Vertex%20Form%20of%20Quadratic%20Equations.webp)
%20How%20to%20Find%20the%20Vertex%20of%20a%20Quadratic%20Function.webp)
%20Transformations%20of%20Quadratic%20Functions%20Using%20the%20Vertex.webp)
Vertex Form of a Quadratic
- The vertex form is , and the vertex is (h, k).
- For example, in , the vertex is (−3, −1).
What Do h and k Mean?
- h moves the graph left or right.
- k moves the graph up or down.
Shifting a Parabola Using Vertex Form
- Start with , which has its vertex at (0,0).
- Moving the vertex 3 left and 1 down gives the new graph .
Practice Questions
Test your understanding
In the vertex form of a quadratic equation, , what does h represent?
Correct! 🎉 +10 pointsNot quite right
h represents the x-value of the vertex, which makes the expression inside the brackets equal to zero. This value shifts the parabola horizontally along the x-axis.
In the vertex form of a quadratic equation, , what does k represent?
Correct! 🎉 +10 pointsNot quite right
k represents the y-value of the vertex, which determines the vertical shift of the parabola.
In the quadratic equation , what is the vertex?
Correct! 🎉 +20 pointsNot quite right
The vertex of the equation is at (4, 7). The term shows that , and the number outside the brackets, 7, represents k.
In the quadratic equation , what is the vertex?
Correct! 🎉 +20 pointsNot quite right
The vertex of the equation is at (−3, −5). The term is equivalent to , so , and .
In the quadratic equation , what is the vertex?
Correct! 🎉 +20 pointsNot quite right
The vertex of the equation is at (1, 6). The term shows that , and the number outside the brackets, 6, represents k.
In the quadratic equation , find the highest point of the parabola.
Correct! 🎉 +30 pointsNot quite right
Since the coefficient of a is negative, the parabola opens downwards, making the vertex the highest point. The term indicates that , and the number outside the brackets, −2, represents k. Therefore, the highest point is at (−5, −2).
Want to see the full working?
Interactive Activity
Vertex form and parabola transformations
Loading interactive widget...
Students Also Ask
The questions students bump into most on this topic
The vertex is the turning point of a parabola, where the curve changes direction. It is the lowest point when the parabola opens upwards and the highest point when it opens downwards. In vertex form, y = a(x - h)² + k, the vertex sits at the point (h, k).
Read the vertex straight from the equation. In y = a(x - h)² + k, the vertex is the point (h, k). Take h from inside the bracket and k from outside it, watching the signs. For example, y = a(x + 3)² - 1 has h = -3 and k = -1.
The value of a tells you which way the parabola opens. When a is positive, the parabola opens upwards and the vertex is its lowest point. When a is negative, it opens downwards and the vertex is its highest point. The value of a does not change when you translate the graph.
Vertex form is written as y = a(x - h)² + k, with a minus before h. So when you see a plus inside the bracket, such as (x + 3), it means x minus negative 3. That makes h equal to -3, because the bracket must be zero at the vertex.
No. Translating a parabola moves it left, right, up or down, which only changes the position of the vertex. The value of a stays the same, so the parabola keeps its shape and opening direction. Only h and k change to record the new position.
In y = a(x - h)² + k, the values h and k give the position of the vertex (h, k). The value h is the horizontal shift, moving the parabola left or right. The value k is the vertical shift, moving it up or down.