2.7 Graphs of Functions

Sketch the graph of y=x2-1{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="space y equals x squared minus 1" style="vertical-align:-6px;height:23px;width:75px" loading="lazy">, labelling any points where the graph intersects the coordinate axes.

Sketch the graph of y=1x{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="space y equals 1 over x" style="vertical-align:-17px;height:47px;width:50px" loading="lazy">, labelling any points where the graph intersects the coordinate axes and stating the equations of any asymptotes.

Sketch the graph of y=x2+3x-4{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="space y equals x squared plus 3 x minus 4" style="vertical-align:-6px;height:23px;width:111px" loading="lazy">, labelling any points where the graph intersects the coordinate axes.

Express2x3+2x2-12x{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="2 x cubed plus 2 x squared minus 12 x" style="vertical-align:-6px;height:23px;width:115px" loading="lazy">in the formax(x+b)(x+c){"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="一个左括号x + b右括号勒ft parenthesis x plus c right parenthesis" style="vertical-align:-6px;height:22px;width:116px" loading="lazy">, wherea, b and c{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="a comma space b space and space c" style="vertical-align:-6px;height:22px;width:72px" loading="lazy">are integers to be found.

Hence sketch the graph of y=2x3+2x2-12x{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="space y equals 2 x cubed plus 2 x squared minus 12 x" style="vertical-align:-6px;height:23px;width:146px" loading="lazy">labelling any points where the graph intersects the coordinate axes.

 y{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="space y" style="vertical-align:-6px;height:22px;width:15px" loading="lazy">is proportional tox{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">.

Whenx=2, y=10{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x equals 2 comma space y equals 10" style="vertical-align:-6px;height:22px;width:91px" loading="lazy">.

通过草图graphs ofy=x3{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="y equals x cubed" style="vertical-align:-6px;height:23px;width:45px" loading="lazy">and y=1x{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="space y equals 1 over x" style="vertical-align:-17px;height:47px;width:50px" loading="lazy">on the same diagram show that there are two real solutions to the equationx3=1x{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="x cubed equals 1 over x" style="vertical-align:-17px;height:47px;width:53px" loading="lazy">.

Use the factor theorem to show that(x-2){"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="left parenthesis x minus 2 right parenthesis" style="vertical-align:-6px;height:22px;width:49px" loading="lazy">is a factor of the function

f(x)=x3-2x2-4x+8{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="straight f left parenthesis x right parenthesis equals x cubed minus 2 x squared minus 4 x plus 8" style="vertical-align:-6px;height:23px;width:170px" loading="lazy">.

Hence, or otherwise, expressf(x){"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="straight f left parenthesis x right parenthesis" style="vertical-align:-6px;height:22px;width:31px" loading="lazy">as a product of three linear factors.

Sketch the graph of y=f(x){"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="space y equals straight f left parenthesis x right parenthesis" style="vertical-align:-6px;height:22px;width:62px" loading="lazy">labelling any points where the graph intersects the coordinate axes.

The diagram below shows the graph of a circle with equation(x-3)2+(y+2)2=9{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="left parenthesis x minus 3 right parenthesis squared plus left parenthesis y plus 2 right parenthesis squared equals 9" style="vertical-align:-6px;height:23px;width:154px" loading="lazy">.

Add straight lines passing through the point to the diagram to show how the circle can have either no, one or two intercepts. All lines must pass through the point(-6,-4){"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}" class="Wirisformula" role="math" alt="left parenthesis negative 6 comma negative 4 right parenthesis" style="vertical-align:-6px;height:22px;width:70px" loading="lazy">.