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Understanding the Impact of Doubling the Value of h on the Graph of y = a(x-h)^2 + k

The graph of a quadratic function in the form of $y = a(x-h)^2 + k$ is a parabola that opens upwards or downwards, depending on the value of $a$. The vertex of the parabola is located at the point $(h, k)$. In this article, we will explore how the graph of this function changes when the value of $h$ is doubled.

The Role of h in the Graph

The value of $h$ in the equation $y = a(x-h)^2 + k$ represents the horizontal translation of the graph. When $h$ is positive, the graph is shifted to the right by $h$ units, and when $h$ is negative, the graph is shifted to the left by $h$ units. The value of $h$ does not affect the shape or the direction of opening of the parabola, but it does affect its position on the $x$-axis.

Doubling the Value of h

When the value of $h$ is doubled, the graph of the function $y = a(x-h)^2 + k$ will also be translated horizontally. However, the direction of the translation will be the same as before, i.e., to the right if $h$ is positive and to the left if $h$ is negative. The magnitude of the translation will be twice the original value of $h$.

The Effect on the Vertex

The vertex of the parabola is located at the point $(h, k)$. When the value of $h$ is doubled, the new vertex will be located at the point $(2h, k)$. This means that the vertex of the graph will move to a point twice as far from the $x$-axis as the original vertex.

The Graph of y = a(x-2h)^2 + k

To visualize the effect of doubling the value of $h$ on the graph, let's consider an example. Suppose we have a parabola with the equation $y = 2(x-2)^2 + 3$. The vertex of this parabola is located at the point $(2, 3)$. If we double the value of $h$, the new equation becomes $y = 2(x-4)^2 + 3$. The vertex of this new parabola is located at the point $(4, 3)$, which is twice as far from the $x$-axis as the original vertex.

In conclusion, when the value of $h$ is doubled in the equation $y = a(x-h)^2 + k$, the graph of the function will also be translated horizontally. The direction of the translation will be the same as before, and the magnitude of the translation will be twice the original value of $h$. The vertex of the parabola will move to a point twice as far from the $x$-axis as the original vertex.

  • The value of $h$ in the equation $y = a(x-h)^2 + k$ represents the horizontal translation of the graph.
  • Doubling the value of $h$ will result in a horizontal translation of twice the original value of $h$.
  • The vertex of the parabola will move to a point twice as far from the $x$-axis as the original vertex.
  • Q: What happens to the graph when the value of $h$ is doubled? A: The graph will be translated horizontally by twice the original value of $h$.
  • Q: How does the vertex of the parabola change when the value of $h$ is doubled? A: The vertex will move to a point twice as far from the $x$-axis as the original vertex.
  • [1] "Graphing Quadratic Functions" by Math Open Reference
  • [2] "Quadratic Functions" by Khan Academy
    Frequently Asked Questions: Understanding the Impact of Doubling the Value of h on the Graph of y = a(x-h)^2 + k

A: The value of h in the equation y = a(x-h)^2 + k represents the horizontal translation of the graph. When h is positive, the graph is shifted to the right by h units, and when h is negative, the graph is shifted to the left by h units.

A: When the value of h is doubled, the graph of the function y = a(x-h)^2 + k will also be translated horizontally. However, the direction of the translation will be the same as before, i.e., to the right if h is positive and to the left if h is negative. The magnitude of the translation will be twice the original value of h.

A: The vertex of the parabola is located at the point (h, k). When the value of h is doubled, the new vertex will be located at the point (2h, k). This means that the vertex of the graph will move to a point twice as far from the x-axis as the original vertex.

A: Suppose we have a parabola with the equation y = 2(x-2)^2 + 3. The vertex of this parabola is located at the point (2, 3). If we double the value of h, the new equation becomes y = 2(x-4)^2 + 3. The vertex of this new parabola is located at the point (4, 3), which is twice as far from the x-axis as the original vertex.

A: The value of h does not affect the shape or the direction of opening of the parabola. However, it does affect its position on the x-axis.

A: Horizontal translation refers to the movement of the graph of a function along the x-axis. In the case of the graph of y = a(x-h)^2 + k, the value of h determines the horizontal translation of the graph. When h is positive, the graph is shifted to the right, and when h is negative, the graph is shifted to the left.

A: The value of a determines the direction and the magnitude of the opening of the parabola. If a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards.

A: Here are the key takeaways:

  • The value of h in the equation y = a(x-h)^2 + k represents the horizontal translation of the graph.
  • Doubling the value of h will result in a horizontal translation of twice the original value of h.
  • The vertex of the parabola will move to a point twice as far from the x-axis as the original vertex.
  • The value of a determines the direction and the magnitude of the opening of the parabola.

A: The graph of y = a(x-h)^2 + k has many real-world applications, including:

  • Modeling the trajectory of a projectile
  • Describing the motion of a pendulum
  • Representing the growth or decay of a population
  • Analyzing the behavior of a quadratic function

A: Here are some resources that you may find helpful:

  • "Graphing Quadratic Functions" by Math Open Reference
  • "Quadratic Functions" by Khan Academy
  • "Algebra II" by Paul Dawkins
  • "Calculus" by Michael Spivak