Question 152607
In order to sketch the graph of a function, we need to know the basic shape. We know that anything with x<sup>2</sup> will be parabola which opens upwards and has its vertex on its bottom. It should have the same basic shape as this graph:<br>
{{{ graph( 300, 200, -5, 5, -5, 5, x^2) }}}
<br>
Now we need to find some points on the graph. Since the problem asks for it, a good place to start would be the vertex. In any equation ax<sup>2</sup> + bx + c, the points of the vertex (h,k) can be found using this equation:
{{{h = -b/2a}}}
<br>Let's plug in the numbers from our equation, x<sup>2</sup> - x - 5
{{{h = -(-1)/2(1)}}}
{{{h = 1/2}}}
<br>Now we can find k by plugging it back into our function:
{{{k = x^2 - x - 5}}}
{{{k = (1/2)^2 - (1/2)-5}}}
{{{k = (1/4) - (1/2) - 5}}}
{{{k = -21/4}}}
<br>Therefore, the vertex is (1/2,-21/4).
<br>Let's find a point on the y-axis this time. The point on the y-axis will have an x = 0:
{{{y = x^2 - x - 5}}}
{{{y = 0^2 - 0 - 5}}}
{{{y = 0 - 0 - 5}}}
{{{y = -5}}}
<br>Now we have a new point: (0,-5)
<br>Since we know that quadratic equations are parallel, we know there has to be another point with the same y value, opposite the same distance from the axis of symmetry of the parabola.
Distance from axis of symmetry: {{{abs(0-(1/2)) = 1/2}}}
<br>Since 0 < 1/2, the other point on the y-axis must be to the left of the axis of symmetry, so lets add twice the distance:
Twice the distance: {{{0 + 2(1/2) = 1}}}
<br>Now we have a new point at (1,-5)
<br>Just to make it easier to graph, let's find some other points on the graph with numbers that are easy to plug into the equation.
<br>If x = 2:
{{{y = 2^2 - 2 - 5}}}
{{{y = 4 - 7}}}
{{{y = -3}}}
<br>As with the y-intercept, we know that there has to be a symmetrical point on the other side of the line of symmetry. Using the same technique:
Distance from axis of symmetry: {{{abs(2-(1/2)) = 3/2}}}
<br>Since 2 > 1/2, the other point must be on the left side of the equation, meaning we need to subtract twice the distance:
Twice the distance: {{{2 - 2(3/2)}}}
 = {{{2 - 3}}}
 = {{{-1}}}
<br>Now we have two more points: (2,-3) and (-1,-3)
<br><br>Let's plot the points we have:<br>
(-1/2,-21/4)
(0,-5)
(1,-5)
(2,-3)
(-1,-3)
<br>
{{{ graph( 300, 200, -5, 5, -6, 6, x^2-x-5) }}}
<br>Be sure to label the vertex at (-1/2, -21/4).