How Does The Graph Of Y = A ( X − H ) 2 + K Y = A(x-h)^2 + K Y = A ( X − H ) 2 + K Change If The Value Of H H H Is Doubled?A. The Vertex Of The Graph Moves To A Point Twice As Far From The X X X -axis.B. The Vertex Of The Graph Moves To A Point Twice As Far From The

Introduction

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 $(h, k)$, which is the minimum or maximum point of the graph. In this article, we will explore how the graph of $y = a(x-h)^2 + k$ changes if the value of $h$ is doubled.

The Effect of Doubling the Value of $h$

When the value of $h$ is doubled, the equation of the graph becomes $y = a(x-2h)^2 + k$. To understand the impact of this change, let's analyze the new equation.

The new equation can be rewritten as $y = a(x^2 - 4hx + 4h^2) + k$. This can be further simplified to $y = ax^2 - 4ahx + 4ah^2 + k$.

Comparing this equation with the original equation $y = a(x-h)^2 + k$, we can see that the coefficient of the $x$ term has changed from $-2ah$ to $-4ah$. This means that the graph has shifted horizontally to the left by a distance of $2h$.

The Vertex of the Graph

The vertex of the graph is located at the point $(h, k)$. When the value of $h$ is doubled, the new vertex is located at the point $(2h, k)$. This means that the vertex of the graph has moved to a point that is twice as far from the $y$-axis.

The Distance of the Vertex from the $x$-axis

The distance of the vertex from the $x$-axis is given by the $y$-coordinate of the vertex, which is $k$. When the value of $h$ is doubled, the new vertex is located at the point $(2h, k)$. This means that the distance of the vertex from the $x$-axis remains the same, which is $k$.

Conclusion

In conclusion, when the value of $h$ is doubled, the graph of $y = a(x-h)^2 + k$ shifts horizontally to the left by a distance of $2h$. The vertex of the graph moves to a point that is twice as far from the $y$-axis, but the distance of the vertex from the $x$-axis remains the same.

Example

Let's consider an example to illustrate this concept. Suppose we have a parabola with the equation $y = (x-2)^2 + 3$. The vertex of this parabola is located at the point $(2, 3)$.

If we double the value of $h$ to $4$, the new equation becomes $y = (x-4)^2 + 3$. The vertex of this new parabola is located at the point $(4, 3)$.

As we can see, the vertex of the new parabola is twice as far from the $y$-axis as the original parabola. However, the distance of the vertex from the $x$-axis remains the same, which is $3$.

Graphical Representation

To visualize the effect of doubling the value of $h$, let's plot the graphs of the two parabolas.

The graph of the original parabola $y = (x-2)^2 + 3$ is a parabola that opens upwards with its vertex at the point $(2, 3)$.

The graph of the new parabola $y = (x-4)^2 + 3$ is a parabola that opens upwards with its vertex at the point $(4, 3)$.

As we can see, the new parabola has shifted horizontally to the left by a distance of $2h$, which is $2$ units in this case.

Conclusion

In conclusion, when the value of $h$ is doubled, the graph of $y = a(x-h)^2 + k$ shifts horizontally to the left by a distance of $2h$. The vertex of the graph moves to a point that is twice as far from the $y$-axis, but the distance of the vertex from the $x$-axis remains the same.

Key Takeaways

• When the value of $h$ is doubled, the graph of $y = a(x-h)^2 + k$ shifts horizontally to the left by a distance of $2h$.
• The vertex of the graph moves to a point that is twice as far from the $y$-axis.
• The distance of the vertex from the $x$-axis remains the same.

Final Thoughts

In this article, we have explored how the graph of $y = a(x-h)^2 + k$ changes when the value of $h$ is doubled. We have seen that the graph shifts horizontally to the left by a distance of $2h$, and the vertex of the graph moves to a point that is twice as far from the $y$-axis. We have also seen that the distance of the vertex from the $x$-axis remains the same.

Q1: What happens to the graph of $y = a(x-h)^2 + k$ when the value of $h$ is doubled?

A1: When the value of $h$ is doubled, the graph of $y = a(x-h)^2 + k$ shifts horizontally to the left by a distance of $2h$. The vertex of the graph moves to a point that is twice as far from the $y$-axis, but the distance of the vertex from the $x$-axis remains the same.

Q2: How does the value of $h$ affect the vertex of the graph?

A2: The value of $h$ affects the location of the vertex of the graph. When the value of $h$ is doubled, the vertex of the graph moves to a point that is twice as far from the $y$-axis.

Q3: What is the effect of doubling the value of $h$ on the distance of the vertex from the $x$-axis?

A3: The distance of the vertex from the $x$-axis remains the same when the value of $h$ is doubled.

Q4: How does the value of $h$ affect the shape of the graph?

A4: The value of $h$ does not affect the shape of the graph. The graph remains a parabola that opens upwards or downwards, depending on the value of $a$.

Q5: Can the value of $h$ be negative?

A5: Yes, the value of $h$ can be negative. In this case, the graph will shift horizontally to the right by a distance of $2h$.

Q6: What happens to the graph of $y = a(x-h)^2 + k$ when the value of $a$ is doubled?

A6: When the value of $a$ is doubled, the graph of $y = a(x-h)^2 + k$ becomes wider or narrower, depending on the value of $h$. The vertex of the graph remains at the same location.

Q7: How does the value of $k$ affect the graph of $y = a(x-h)^2 + k$?

A7: The value of $k$ affects the vertical position of the graph. The graph is shifted upwards or downwards by a distance of $k$.

Q8: Can the value of $k$ be negative?

A8: Yes, the value of $k$ can be negative. In this case, the graph will be shifted downwards by a distance of $k$.

Q9: What is the effect of doubling the value of $k$ on the graph of $y = a(x-h)^2 + k$?

A9: When the value of $k$ is doubled, the graph of $y = a(x-h)^2 + k$ is shifted upwards by a distance of $2k$.

Q10: How can I visualize the graph of $y = a(x-h)^2 + k$?

A10: You can visualize the graph of $y = a(x-h)^2 + k$ by plotting the equation on a coordinate plane. You can use graphing software or a calculator to plot the graph.

Conclusion

In this article, we have answered some frequently asked questions about the graph of $y = a(x-h)^2 + k$. We have seen how the value of $h$ affects the location of the vertex of the graph, and how the value of $k$ affects the vertical position of the graph. We have also seen how the value of $a$ affects the shape of the graph.