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We will now study the graphs of quadratic equations in two variables with general form $$\begin{array}{lllll}y=a{x}^{2}+bx+c,\hfill & \hfill & a\ne 0,\hfill & \hfill & a,b,c\text{\hspace{0.17em}}\text{are\hspace{0.17em}real\hspace{0.17em}numbers}\hfill \end{array}$$
We will construct the graph of a parabola by choosing several $x$ -values, computing to find the corresponding $y$ -values, plotting these ordered pairs, then drawing a smooth curve through them.
Graph
$y={x}^{2}.$ Construct a table to exhibit several ordered pairs.
$x$ | $y={x}^{2}$ |
0 | 0 |
1 | 1 |
2 | 4 |
3 | 9 |
$-1$ | 1 |
$-2$ | 4 |
$-3$ | 9 |
Graph
$y={x}^{2}-2.$ Construct a table of ordered pairs.
$x$ | $y={x}^{2}-2$ |
0 | $-2$ |
1 | $-1$ |
2 | 2 |
3 | 7 |
$-1$ | $-1$ |
$-2$ | 2 |
$-3$ | 7 |
Use the idea suggested in Sample Set A to sketch (quickly and perhaps not perfectly accurately) the graphs of
$$\begin{array}{lllll}y={x}^{2}+1\hfill & \hfill & \text{and}\hfill & \hfill & y={x}^{2}-3\hfill \end{array}$$
Graph
$y={\left(x+2\right)}^{2}.$
Do we expect the graph to be similar to the graph of
$y={x}^{2}$ ? Make a table of ordered pairs.
$x$ | $y$ |
0 | 4 |
1 | 9 |
$-1$ | 1 |
$-2$ | 0 |
$-3$ | 1 |
$-4$ | 4 |
Use the idea suggested in Sample Set B to sketch the graphs of
$$\begin{array}{lllll}y={\left(x-3\right)}^{2}\hfill & \hfill & \text{and}\hfill & \hfill & y={\left(x+1\right)}^{2}\hfill \end{array}$$
For the following problems, graph the quadratic equations.
$y=-{x}^{2}$
$y={\left(x-2\right)}^{2}$
$y={\left(x+1\right)}^{2}$
$y={x}^{2}-1$
$y={x}^{2}+\frac{1}{2}$
$y=-{x}^{2}+1$ (Compare with problem 2.)
$y={\left(x-1\right)}^{2}-1$
$y=-{\left(x+1\right)}^{2}$
$y=2{x}^{2}$
$y=\frac{1}{2}{x}^{2}$
For the following problems, try to guess the quadratic equation that corresponds to the given graph.
( [link] ) Simplify and write ${\left({x}^{-4}{y}^{5}\right)}^{-3}{\left({x}^{-6}{y}^{4}\right)}^{2}$ so that only positive exponents appear.
( [link] ) Factor ${y}^{2}-y-42.$
$\left(y+6\right)\left(y-7\right)$
( [link] ) Find the sum: $\frac{2}{a-3}+\frac{3}{a+3}+\frac{18}{{a}^{2}-9}.$
( [link] ) Simplify $\frac{2}{4+\sqrt{5}}.$
$\frac{8-2\sqrt{5}}{11}$
( [link] ) Four is added to an integer and that sum is doubled. When this result is multiplied by the original integer, the product is $-6.$ Find the integer.
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