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Chapter 0: Problem 5

Graph each pair of equations on one set of axes. $$y=-2 x^{2} \text { and } y=-2 x^{2}+1$$

Short Answer

Expert verified
Graph \(y = -2x^2\) and \(y = -2x^2 + 1\). The first parabola is at \((0,0)\) and the second is at \((0,1)\).

Step by step solution

01

Understand the equations

Identify the given equations. The first equation is \(y = -2x^2\). The second equation is \(y = -2x^2 + 1\). Notice that these are both quadratic equations, and one is a simple vertical shift of the other.
02

Identify the vertex of each parabola

For the equation \(y = -2x^2\), the vertex is at the origin \((0,0)\). For the equation \(y = -2x^2 + 1\), the vertex is shifted up by 1 unit, so the vertex is at \((0, 1)\).
03

Determine the shape and direction of the parabolas

Both equations have the same coefficient for the \(x^2\) term, \(-2\), which means they open downward and are stretched vertically by a factor of \2\.
04

Plot key points

Choose key points to plot for each curve. For \(y = -2x^2\), plot points such as \((0,0)\), \((1, -2)\), and \((-1, -2)\). For \(y = -2x^2 + 1\), plot \((0,1)\), \((1, -1)\), and \((-1, -1)\).
05

Draw the parabolas

Graph the points and sketch each parabola. The parabola for \(y = -2x^2\) should pass through \((0,0)\), \((1,-2)\), and \((-1, -2)\). The parabola for \(y = -2x^2 + 1\) should pass through \((0,1)\), \((1,-1)\), and \((-1, -1)\).

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Key Concepts

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

quadratic equations
Quadratic equations are polynomial equations of the form \(y = ax^2 + bx + c\). These equations are called quadratic because 'quad' means square, referring to the square of the variable \(x\). For the given equations, \(y = -2x^2\) and \(y = -2x^2 + 1\), notice the \(x^2\) term.
The 'a' coefficient tells us about the direction and shape of the parabola:
  • If 'a' is positive, the parabola opens upwards.
  • If 'a' is negative, the parabola opens downwards.

Here, both equations have \(a = -2\), so both parabolas open downward. The 'b' and 'c' coefficients change the position and shape depending on their values. But our equations lack an 'x' term (b=0), simplifying things.
parabolas
A parabola is the graph of a quadratic equation. It's a symmetric, U-shaped curve. For the equations \(y = -2x^2\) and \(y = -2x^2 + 1\), they form parabolas.
Key properties include:
  • The vertex, the highest or lowest point (turning point).
  • The axis of symmetry which passes through the vertex and divides the parabola into two mirror images.
  • The direction in which it opens (up or down) determined by the sign of 'a'.

Since our 'a' is negative, both parabolas open downward. The vertex's y-coordinate shows the maximum point.
vertex shift
The vertex shift is how the vertex of the parabola moves from its typical position. For the standard quadratic equation \(y = ax^2\), the vertex is at (0,0). In our second equation, \(y = -2x^2 + 1\), the +1 means a vertical shift.
  • When a constant is added (e.g., +1), the vertex moves up.
  • When a constant is subtracted (e.g., -1), the vertex moves down.

In the first equation, the vertex is at (0,0). In the second, it's at (0,1). Just add the constant to the original y-coordinate to find the new vertex position.
plotting points
Plotting key points helps sketch accurate graphs of parabolas. Choose symmetrical points around the vertex. Here鈥檚 how:
  • Start with the vertex. For \(y = -2x^2\), it's (0,0). For \(y = -2x^2 + 1\), it's (0,1).
  • Choose points on either side. For example, for \(y = -2x^2\), points like (1,-2) and (-1,-2).
  • Do the same for the second equation. For \(y = -2x^2 + 1\), choose points like (1,-1) and (-1,-1).

Plot these points on your graph, then draw smooth curves connecting the points to form the parabolas. This method ensures accuracy and symmetry.

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Most popular questions from this chapter

Quick Copy buys an office machine for 5200 dollars on January 1 of a given year. The machine is expected to last for 8 yr, at the end of which time its salvage value will be 1100 dollars. If the company figures the decline in value to be the same each year, then the book value, \(V(t),\) after \(t\) years, \(0 \leq t \leq 8,\) is given by $$V(t)=C-t\left(\frac{C-S}{N}\right)$$ where \(C\) is the original cost of the item, \(N\) is the number of years of expected life, and \(S\) is the salvage value. a) Find the linear function for the straight-line depreciation of the office machine. b) Find the book value after 0 yr, 1 yr, 2 yr, 3 yr, 4 yr, 7 yr, and 8 yr.

Find the domain of each function given below. $$g(x)=\sqrt{4+5 x}$$

Solve for \(y\) in terms of \(x\). Decide whether the resulting equation represents a function. $$\left(3 y^{3 / 2}\right)^{2}=72 x$$

Find the domain of each function given below. $$g(x)=|x|+1$$

In January 2001, 3300 manatees were counted in an aerial survey of Florida. In January 2005,3143 manatees were counted. (Source: Florida Fish and Wildlife Conservation Commission.) a) Using the year as the \(x\) -coordinate and the number of manatees as the \(y\) -coordinate, find an equation of the line that contains the two data points. b) Use the equation in part (a) to estimate the number of manatees counted in January 2010. c) The actual number counted in January 2010 was 5067 . Does the equation found in part (a) give an accurate representation of the number of manatees counted each year?
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