12 Integrated Applications Free Online FlashCards

What is a reliable strategy for solving cumulative applied problems?

Define variables with units, identify the relationship, choose a useful form, determine parameters from constraints, solve symbolically, check the result, and interpret it in context.

When is a linear model appropriate?

Use a linear model when a quantity changes by approximately a constant amount per unit of input: $f(x)=mx+b$.

How are $m$ and $b$ found from two points?

For points $(x_1,y_1)$ and $(x_2,y_2)$, the slope is $m=\frac{y_2-y_1}{x_2-x_1}$, and the intercept is $b=y_1-mx_1$.

What does vertex form reveal about a parabola?

In $y=a(x-h)^2+k$, the vertex is $(h,k)$. The sign of $a$ shows whether the parabola opens upward or downward.

How is the period found in a sinusoidal model?

For $y=A\sin(Bx-C)+D$ or $y=A\cos(Bx-C)+D$, the period is $P=\frac{2\pi}{|B|}$.

How is a percentage rate converted to an exponential factor?

For percentage growth, use $b=1+r$; for percentage decay, use $b=1-r$ in $P(t)=P_0b^t$.

How are logarithmic and exponential forms related?

The statement $\log_b(x)=y$ is equivalent to $b^y=x$. A real logarithm requires a positive argument.

What formula models an arithmetic sequence?

An arithmetic sequence has a constant difference and uses $a_n=a_1+(n-1)d$.

What formula models a geometric sequence?

A geometric sequence has a constant ratio and uses $a_n=a_1r^{n-1}$.

Which trigonometric ratio compares opposite and adjacent sides?

For a right triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Thus an angle can be found with inverse tangent when the two legs are known.

What is the Law of Cosines?

The Law of Cosines is $c^2=a^2+b^2-2ab\cos C$, relating two sides and their included angle to the opposite side.

What is the drone path’s maximum point?

For $y=-0.02(x-30)^2+18$, the vertex is $(30,18)$, so the modeled maximum height is $18$ meters at $x=30$ meters.