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Chapter 1: Problem 49

Graph the function using transformations. $$y=2 x^{2}+1$$

Short Answer

Expert verified
Graph the function by stretching \( y=x^2 \) vertically by 2, and moving it up by 1 unit.

Step by step solution

01

Start with the Parent Function

The parent function of \( y = 2x^2 + 1 \) is \( y = x^2 \). This is a basic quadratic function that has a U-shaped graph called a parabola. The vertex of this parent function is at the origin (0,0), and it opens upwards.
02

Apply Vertical Stretch

The coefficient '2' in \( y = 2x^2 + 1 \) indicates a vertical stretch by a factor of 2. Each point on the parent function is moved twice as far from the x-axis. For example, the point (1,1) on \( y = x^2 \) will become (1,2) on \( y = 2x^2 \).
03

Apply Vertical Translation

The '+1' in \( y = 2x^2 + 1 \) translates the graph of the function upward by 1 unit. This means each point on the stretched parabola moves 1 unit up. So, the vertex moves from (0,0) to (0,1).
04

Graph the Transformed Function

Now, using the transformations from Steps 2 and 3, we graph the function. The new parabola has its vertex at (0,1), is vertically stretched by a factor of 2, and opens upwards. Points like (1,1) on \( y = x^2 \) are now (1,3) on \( y = 2x^2 + 1 \) after applying both transformations.

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Key Concepts

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

Understanding Quadratic Functions
Quadratic functions are a type of polynomial function and are known for their characteristic U-shaped graph called a parabola. The most basic form of a quadratic function is \( y = x^2 \). This function involves a variable raised to the second power, hence the name "quadratic." Quadratics are often used as parent functions in algebra to explore various transformations of the graph. The standard form of a quadratic equation is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.
  • When graphed, the parabola of any quadratic function will have a vertex, which is the peak or the lowest point of the graph, depending on whether it opens upwards or downwards.
  • Quadratic functions have a symmetrical nature, meaning if you were to fold the graph along the line that passes through its vertex, both sides would match.
  • The axis of symmetry is a vertical line through the vertex that divides the parabola into two symmetrical halves.
What is a Parabola?
In mathematics, a parabola is a specific type of curve on a graph formed by a quadratic function like \( y = x^2 \). Imagine it as a smooth, symmetrical U-shape. Parabolas can open upwards or downwards based on the function's expression. Here's how you understand the parabola:
  • If the coefficient \( a \) in \( y = ax^2 + bx + c \) is positive, the parabola opens upwards, making a smiley face.
  • If \( a \) is negative, it opens downward, resembling a frown.
  • The vertex of the parabola is the highest or lowest point, and it is a critical feature of the graph. The vertex helps in identifying the maximum or minimum value of the quadratic function.
Parabolas are quite versatile; they can model real-world situations like projectile motion and are significant in fields such as physics and engineering.
Exploring Vertical Stretch
A vertical stretch occurs in a quadratic function when the coefficient of \( x^2 \) changes from its parent value. In the function \( y = 2x^2 \), the coefficient of 2 causes every point on the graph of the parent function \( y = x^2 \) to move twice as far away from the x-axis.
  • This makes the parabola appear narrower compared to its parent function.
  • The vertical stretch can be thought of as a pulling action that elongates the curve along the y-axis.
For instance, any point \((x, y)\) on the parabola \( y = x^2 \) is transformed to \((x, 2y)\) in \( y = 2x^2 \). A vertical stretch affects how steep or flat the parabola appears, significantly altering its visual form on a graph.
Understanding Vertical Translation
Vertical translation is a type of transformation that moves a graph up or down the coordinate plane. Looking at the function \( y = x^2 + 1 \), the \(+1\) indicates a vertical translation up by 1 unit, meaning the entire parabola shifts upward.
  • This transformation does not affect the 'width' or 'narrowness' of the parabola, only its vertical position.
  • The vertex, which was initially at (0,0) for \( y = x^2 \), is shifted to (0,1) with the vertical translation.
Such translations are easy to identify as they visibly alter the starting point of the graph without changing its overall shape. Vertical translations are straightforward but have powerful implications on positioning the function relative to the x-axis.

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Most popular questions from this chapter

In Exercises \(95-98,\) explain the mistake that is made. Most money market accounts pay a higher interest with a higher principal. If the credit union is offering \(2 \%\) on accounts with less than or equal to \(\$ 10,000\) and \(4 \%\) on the additional money over \(\$ 10,000,\) write the interest function \(I(x)\) that represents the interest earned on an account as a function of dollars in the account. Solution: \(I(x)=\left\\{\begin{array}{ll}0.02 x & x \leq 10,000 \\\ 0.02(10,000)+0.04 x & x>10,000\end{array}\right.\) This is incorrect. What mistake was made?

A \(20 \mathrm{ft} \times 10\) ft rectangular pool has been built. If 50 cubic feet of water is pumped into the pool per hour, write the water-level height (feet) as a function of time (hours).

In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=\sqrt[3]{8 x-1}, \quad g(x)=\frac{x^{3}+1}{8}$$

In Exercises \(83-86,\) determine whether each statement is true or false. The domain of a composite function can be found by inspection, without knowledge of the domain of the individual functions.

Determine whether each statement is true or false. If \(g(x)=\frac{1}{b-x}\) and \(g(3)\) is undefined, find \(b\)
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