Linear and Nonlinear Functions 03:50 minutes

Video Transcript

Transcript Linear and Nonlinear Functions

Welcome to the 65th annual Pole Position Championship! Where we have AIR TON from team Senna vs. Elaine from team Prost. Elaine seems really relaxed ahead of the final race of the season while AIR TON is carefully examining the track.

Linear and non-linear Functions

The race starts with a linear track where the racers can build up speed. Then, they have to transition to the quadratic portion of the track where they whip around Parabolica’s curve. The final leg of the race is the hardest. The cubic portion of the track requires the racers to navigate the tricky s-curve before they race to the checkered flag.

Linear Functions

To understand the track better, let's look at linear and non-linear functions. AIR TON’s team thinks it’d be a good idea if she has a different strategy for each portion of the race, so they analyze each part separately. The linear portion of the track is shown here.
As AIR TON knows, when linear functions are graphed, they always look like a straight line and are in the form f(x) = mx + b. Remember, ‘m’ is the slope and ‘b’ is the y-intercept. If ‘m’ is positive, the graph of the line increases from left to right. If ‘m’ is negative, the graph of the line decreases from left to right. So just remember, each one of these lines is also a linear function. The positive slopes generally look like this and the negative slopes generally look like this. This particular track has the equation f(x) = 4x + 22.

Quadratic Functions

The next part of the track AIR TON needs to analyze is Parabolica’s curve. Parabolica, that sounds like ‘parabola’. Quadratic functions, like the one shown here, either open down or open up. The functions are in the form f(x) = ax² + bx + c. If ‘a’ is positive, the graph opens upwards and downwards if ‘a’ is negative. Notice how the graph of quadratic functions have one ‘bend’ in the graph. This particular part of the track has the equation g(x) = -x² - 6x - 3. Notice how 'a' is negative and the graph opens downwards.

Cubic Functions

The last part of the track is Cubic Curve. Cubic functions have the form f(x) = ax³ + bx² + cx + d. If ‘a’ is positive, the graph of the line increases from left to right. If ‘a’ is negative, the graph of the line decreases from left to right. The graph of cubic functions have two ‘bends’ in the graph. So quadratic functions have one bend and cubic functions have two bends. The final curve has the expression h(x) = -1.5x³ + 3x² + 2x - 1

Let’s rejoin the racers at the starting line. Aaaaaand they’re off. well, one of them anyway.