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Chapter 1: Problem 12

How do you obtain the graph of \(y=4(x+3)^{2}+6\) from the graph of \(y=x^{2} ?\)

Short Answer

Expert verified
Question: Identify the transformations needed to obtain the graph of \(y=4(x+3)^2+6\) from the graph of \(y=x^2\) and provide a step by step solution. Answer: To obtain the graph of \(y=4(x+3)^2+6\) from the graph of \(y=x^2\), we need to perform the following transformations: 1. Apply a vertical stretch by a factor of 4. 2. Apply a horizontal shift to the left by 3 units. 3. Apply a vertical shift up by 6 units.

Step by step solution

01

Identify the transformations in the equation

In the given equation \(y = 4(x+3)^2 + 6\), we can see the following transformations: 1. The coefficient \(4\) outside the squared term is a vertical stretch by a factor of 4. 2. The term \((x+3)\) inside the squared term corresponds to a horizontal shift to the left by 3 units. 3. The constant term \(+6\) added to the equation corresponds to a vertical shift up by 6 units. Now that we have identified the transformations, we can proceed to apply them step by step.
02

Apply vertical stretch

We begin by applying a vertical stretch by factor 4 to the graph of \(y=x^2\). This stretch makes the parabola narrower, with the same vertex. The equation of the graph after the vertical stretch becomes \(y=4x^2\).
03

Apply horizontal shift

Next, we apply a horizontal shift to the left by 3 units on the graph of \(y=4x^2\). The vertex of the parabola moves from \((0,0)\) to \((-3,0)\). The equation of the graph after the horizontal shift is \(y=4(x+3)^2\).
04

Apply vertical shift

Finally, we apply a vertical shift up by 6 units to the graph of \(y=4(x+3)^2\). The vertex of the parabola moves from \((-3,0)\) to \((-3,6)\). The final equation of the graph is \(y=4(x+3)^2+6\), as required. To summarize, to obtain the graph of \(y=4(x+3)^2+6\) from the graph of \(y=x^2\), we need to perform the following transformations: 1. Apply a vertical stretch by a factor of 4. 2. Apply a horizontal shift to the left by 3 units. 3. Apply a vertical shift up by 6 units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Stretch
Imagine the familiar parabola of \(y=x^2\), gently opening up like a bowl. When we apply a vertical stretch, this shape becomes narrower. It is as if we grabbed the top of the parabola and pulled it upward, without changing the position of the vertex. This transformation involves multiplying the whole quadratic expression by a constant factor. For our equation \(y=4(x+3)^2+6\), the constant factor is \(4\). A vertical stretch by this factor scales all \(y\)-values by \(4\). If a point on the original graph was at \((x, y)\), after the vertical stretch it moves to \((x, 4y)\). The effect is that the parabola appears steeper. Key Points:
  • Vertical stretch changes the steepness of the parabola.
  • The vertex remains unaffected during this transformation.
  • It impacts the shape but not the orientation of the graph.
Horizontal Shift
A horizontal shift slides the entire graph left or right across the plane. In the quadratic equation \(y=4(x+3)^2+6\), the horizontal shift is represented by \(x+3\). This means the graph of \(y=4x^2\) is shifted left by \(3\) units. Our starting point before this shift is the vertex, which originally is at \((0,0)\). To perform this shift, replace \(x\) with \(x+3\) in the equation. Now the vertex at the origin moves to \((-3,0)\). This transformation changes the position of the graph on the x-axis.What to Remember:
  • Always pay attention to the sign: \((x+3)\) shifts the graph left, whereas \((x-3)\) would shift it right.
  • Horizontal shifts do not affect the shape of the graph.
  • It鈥檚 like sliding the whole function sideways on the graph.
Vertical Shift
A vertical shift moves the graph up or down without altering its shape. This shift is indicated by the constant term \(+6\) in our quadratic equation \(y=4(x+3)^2+6\). Adding \(6\) to the equation lifts the entire graph upward by \(6\) units. After the vertical stretch and horizontal shift, our vertex was at \((-3,0)\). With a vertical shift, it moves to \((-3,6)\). This means every point on the graph is raised equally along the y-axis.Pointers for Vertical Shifts:
  • Both up and down shifts affect the entire graph uniformly.
  • The shift does not change the width or orientation of the parabola.
  • Think of it as purely adjusting the "altitude" of the graph.

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