Interactive practice questions
The function $y=ax^2-bx+c$y=ax2−bx+c passes through the points ($5$5, $-42$−42) and ($4$4, $-66$−66) and has a maximum turning point at $x=3$x=3.
Form an equation by substituting ($5$5, $-42$−42) into the function.
Form another equation by substituting ($4$4, $-66$−66) into the function.
Find $\frac{dy}{dx}$dydx.
Form an equation by using the fact that the function has a maximum turning point at $x=3$x=3.
Make $b$b the subject of the equation.
Substitute $b=6a$b=6a into Equation 1.
| Equation 1 | $-42=25a-5b+c$−42=25a−5b+c |
| Equation 2 | $-66=16a-4b+c$−66=16a−4b+c |
Substitute $b=6a$b=6a into Equation 2.
| Equation 1 | $-42=25a-5b+c$−42=25a−5b+c |
| Equation 2 | $-66=16a-4b+c$−66=16a−4b+c |
Solve for $a$a.
| Equation 1 | $-42=-5a+c$−42=−5a+c |
| Equation 2 | $-66=-8a+c$−66=−8a+c |
Solve for $c$c.
| Equation 1 | $-42=-5a+c$−42=−5a+c |
| Equation 2 | $-66=-8a+c$−66=−8a+c |
Find the value of $b$b.
The function $f\left(x\right)=ax^2+\frac{b}{x^2}$f(x)=ax2+bx2 has turning points at $x=1$x=1 and $x=-1$x=−1.
Consider the function $y=x^3-ax^2+bx+11$y=x3−ax2+bx+11.
The function $f\left(x\right)=ax^3+18x^2+cx+4$f(x)=ax3+18x2+cx+4 has turning points at $x=4$x=4 and $x=2$x=2.