Question 335943
Finding the x-intercepts:



Simply plug in {{{f(x)=0}}} to get {{{0=x^2-2x-4}}}. The goal now is to solve for 'x'.



Notice that the quadratic {{{x^2-2x-4}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=1}}}, {{{B=-2}}}, and {{{C=-4}}}



Let's use the quadratic formula to solve for "x":



{{{x = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{x = (-(-2) +- sqrt( (-2)^2-4(1)(-4) ))/(2(1))}}} Plug in  {{{A=1}}}, {{{B=-2}}}, and {{{C=-4}}}



{{{x = (2 +- sqrt( (-2)^2-4(1)(-4) ))/(2(1))}}} Negate {{{-2}}} to get {{{2}}}. 



{{{x = (2 +- sqrt( 4-4(1)(-4) ))/(2(1))}}} Square {{{-2}}} to get {{{4}}}. 



{{{x = (2 +- sqrt( 4--16 ))/(2(1))}}} Multiply {{{4(1)(-4)}}} to get {{{-16}}}



{{{x = (2 +- sqrt( 4+16 ))/(2(1))}}} Rewrite {{{sqrt(4--16)}}} as {{{sqrt(4+16)}}}



{{{x = (2 +- sqrt( 20 ))/(2(1))}}} Add {{{4}}} to {{{16}}} to get {{{20}}}



{{{x = (2 +- sqrt( 20 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (2 +- 2*sqrt(5))/(2)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{x = (2)/(2) +- (2*sqrt(5))/(2)}}} Break up the fraction.  



{{{x = 1 +- 1*sqrt(5)}}} Reduce.  



{{{x = 1+1*sqrt(5)}}} or {{{x = 1-1*sqrt(5)}}} Break up the expression.  



So the solutions are {{{x = 1+1*sqrt(5)}}} or {{{x = 1-1*sqrt(5)}}} 



which approximate to {{{x=3.236}}} or {{{x=-1.236}}} 



So the x-intercepts are roughly (3.236, 0) and (-1.236, 0)



In order to graph {{{y=x^2-2x-4}}}, we need to plot some points.



Let's find y when {{{x=-2}}}

(Note: you can start with any x-value):



{{{y=x^2-2x-4}}} Start with the given equation.



{{{y=(-2)^2-2(-2)-4}}} Plug in {{{x=-2}}}.



{{{y=4-2(-2)-4}}} Square {{{-2}}} to get {{{4}}}.



{{{y=4+4-4}}} Multiply {{{-2}}} and {{{-2}}} to get {{{4}}}.



{{{y=4}}} Combine like terms.



So when {{{x=-2}}}, then {{{y=4}}}.



So we have the point *[Tex \LARGE \left(-2,4\right)].



----------------------------------------------------



Let's find y when {{{x=-1}}}:



{{{y=x^2-2x-4}}} Start with the given equation.



{{{y=(-1)^2-2(-1)-4}}} Plug in {{{x=-1}}}.



{{{y=1-2(-1)-4}}} Square {{{-1}}} to get {{{1}}}.



{{{y=1+2-4}}} Multiply {{{-2}}} and {{{-1}}} to get {{{2}}}.



{{{y=-1}}} Combine like terms.



So when {{{x=-1}}}, then {{{y=-1}}}.



So we have the point *[Tex \LARGE \left(-1,-1\right)].



----------------------------------------------------



Let's find y when {{{x=0}}}:



{{{y=x^2-2x-4}}} Start with the given equation.



{{{y=(0)^2-2(0)-4}}} Plug in {{{x=0}}}.



{{{y=0-2(0)-4}}} Square {{{0}}} to get {{{0}}}.



{{{y=0+0-4}}} Multiply {{{-2}}} and {{{0}}} to get {{{0}}}.



{{{y=-4}}} Combine like terms.



So when {{{x=0}}}, then {{{y=-4}}}.



So we have the point *[Tex \LARGE \left(0,-4\right)].



----------------------------------------------------



Let's find y when {{{x=1}}}:



{{{y=x^2-2x-4}}} Start with the given equation.



{{{y=(1)^2-2(1)-4}}} Plug in {{{x=1}}}.



{{{y=1-2(1)-4}}} Square {{{1}}} to get {{{1}}}.



{{{y=1-2-4}}} Multiply {{{-2}}} and {{{1}}} to get {{{-2}}}.



{{{y=-5}}} Combine like terms.



So when {{{x=1}}}, then {{{y=-5}}}.



So we have the point *[Tex \LARGE \left(1,-5\right)].



----------------------------------------------------



Let's find y when {{{x=2}}}:



{{{y=x^2-2x-4}}} Start with the given equation.



{{{y=(2)^2-2(2)-4}}} Plug in {{{x=2}}}.



{{{y=4-2(2)-4}}} Square {{{2}}} to get {{{4}}}.



{{{y=4-4-4}}} Multiply {{{-2}}} and {{{2}}} to get {{{-4}}}.



{{{y=-4}}} Combine like terms.



So when {{{x=2}}}, then {{{y=-4}}}.



So we have the point *[Tex \LARGE \left(2,-4\right)].



----------------------------------------------------



Let's find y when {{{x=3}}}:



{{{y=x^2-2x-4}}} Start with the given equation.



{{{y=(3)^2-2(3)-4}}} Plug in {{{x=3}}}.



{{{y=9-2(3)-4}}} Square {{{3}}} to get {{{9}}}.



{{{y=9-6-4}}} Multiply {{{-2}}} and {{{3}}} to get {{{-6}}}.



{{{y=-1}}} Combine like terms.



So when {{{x=3}}}, then {{{y=-1}}}.



So we have the point *[Tex \LARGE \left(3,-1\right)].



----------------------------------------------------



Let's find y when {{{x=4}}}:



{{{y=x^2-2x-4}}} Start with the given equation.



{{{y=(4)^2-2(4)-4}}} Plug in {{{x=4}}}.



{{{y=16-2(4)-4}}} Square {{{4}}} to get {{{16}}}.



{{{y=16-8-4}}} Multiply {{{-2}}} and {{{4}}} to get {{{-8}}}.



{{{y=4}}} Combine like terms.



So when {{{x=4}}}, then {{{y=4}}}.



So we have the point *[Tex \LARGE \left(4,4\right)].



----------------------------------------------------


Now let's make a table of the values we just found.



<h4>Table of Values:</h4><pre>

<TABLE border="1" width="100">
<TR><TD>x</TD><TD>y</TD></TR><tr><td>-2</td><td>4</td></tr>
<tr><td>-1</td><td>-1</td></tr>
<tr><td>0</td><td>-4</td></tr>
<tr><td>1</td><td>-5</td></tr>
<tr><td>2</td><td>-4</td></tr>
<tr><td>3</td><td>-1</td></tr>
<tr><td>4</td><td>4</td></tr>
</TABLE>

</pre>

Now let's plot the points:



{{{ drawing(500, 500, -10,10,-10,10,
grid(1),
graph(500, 500, -10,10,-10,10, 0),
circle(-2,4,0.08),circle(-2,4,0.10),
circle(-1,-1,0.08),circle(-1,-1,0.10),
circle(0,-4,0.08),circle(0,-4,0.10),
circle(1,-5,0.08),circle(1,-5,0.10),
circle(2,-4,0.08),circle(2,-4,0.10),
circle(3,-1,0.08),circle(3,-1,0.10),
circle(4,4,0.08),circle(4,4,0.10)

)}}}


<h4>Graph:</h4>

Now draw a curve through all of the points to graph {{{y=x^2-2x-4}}}:



{{{ drawing(500, 500, -10,10,-10,10,
grid(1),
graph(500, 500, -10,10,-10,10, x^2-2x-4),
circle(-2,4,0.08),circle(-2,4,0.10),
circle(-1,-1,0.08),circle(-1,-1,0.10),
circle(0,-4,0.08),circle(0,-4,0.10),
circle(1,-5,0.08),circle(1,-5,0.10),
circle(2,-4,0.08),circle(2,-4,0.10),
circle(3,-1,0.08),circle(3,-1,0.10),
circle(4,4,0.08),circle(4,4,0.10)

)}}} Graph of {{{y=x^2-2x-4}}}



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Jim