Nonlinear functions

A nonlinear function has a graph that isn't a straight line. A quadratic has an $x^2$ term or two factors like $(x-4)(x+2)$, and its graph is a parabolaThe U-shaped curve a quadratic draws, opening either up or down.. An exponential has the variableA letter that stands for a number you don’t know yet, like the x in 5x + 7.Read the lesson in the exponentThe small raised number that says how many times to multiply, like the 3 in x cubed.Read the lesson, as in $480(1.15)^w$, and it shows up whenever a story repeats a percent change. You'll also see cubes and other polynomialsAn expression built from powers of one letter added together, like x cubed minus 2x plus 5.Read the lesson, absolute valueHow far a number is from zero, so it’s never negative, like 5 for both 5 and -5.Read the lesson bars, and a variable in a denominatorThe bottom number of a fraction, like the 4 in 3/4.Read the lesson.

Read the last line first. It might say:

• “What is the value of $f(x)$ when $x=5$?”
• “What is the vertex of the graph?”
• “Which equation represents this situation?”
• “Which of the following is the best interpretation of 480 in this context?”
• “Which of the following equations displays, as a constantA number that doesn’t change, like the 7 in 5x + 7, or a letter such as p that stands for one fixed value.Read the lesson or coefficientThe number multiplying a letter, like the 5 in 5x.Read the lesson, the minimum value of the function it defines?”

Desmos is fast when you're given a function and asked for a number. When the question asks what a number means or which form shows a feature, or the graph is already printed, leave it closed. The math guide covers typing a grid-inA question where you type your own answer instead of picking one of four.Read the lesson answer.

In function notationWriting a function as f(x), where the number in the parentheses is the one going in.Read the lesson, the number in the parentheses is the input. For $f(x)=\frac{12}{x+1}$, $f(5)=\frac{12}{5+1}=2$.

When the question gives you the output instead, set the whole formula equal to it. Say $k(t)=4t^3$, and the question asks for $t$ when $k(t)=500$.

1. Set the formula equal to the output.

   $$4t^3=500$$

2. Divide both sides by 4.

   $$t^3=125$$

3. Take the cube root: $5\cdot5\cdot5=125$.

   $$t=5$$

On a table with evenly spaced inputs, compare the outputs two ways. Subtract to get the changes, and divide to get the ratiosA comparison of two amounts, like 3 to 5.Read the lesson.

Check the spacing first. If the inputs go 0, 2, 4, each change or ratio covers two steps.

On a graph, a parabola turns at its vertex. A shrinking exponential like $80(0.5)^x$ never turns. It flattens toward the horizontal axis without reaching it, so it has no lowest point.

The y-interceptThe output when the input is 0, which is where a graph crosses the vertical axis.Read the lesson is the output when the input is 0. Put in 0, and every term with an $x$ in it drops out.

A zero of a function is an input that gives an output of 0. On the graph, it's an x-interceptThe value of x where a graph crosses the horizontal axis, so the output there is 0.Read the lesson. Factored formAn expression written as factors multiplied together, like (x - 2)(x + 5), where setting each factor to zero gives a solution.Read the lesson, $f(x)=a(x-p)(x-q)$, shows the zeros once you flip each sign.

It works backward too. A cubic with zeros $-4$, 1 and 3 is $f(x)=a(x+4)(x-1)(x-3)$, since each factor is $x$ minus a zero. $a(x-4)(x+1)(x+3)$ has every sign backward.

A parabola has two mirror-image halves. The vertical line between them runs through the vertex, and it's called the axis of symmetryThe vertical line through a parabola’s vertex that splits it into two mirror-image halves.. Find that line, then put its $x$ back into the function for the other coordinate.

For $f(x)=x^2-6x+14$, $a=1$ and $b=-6$.

1. Put $a$ and $b$ into the formula, signs included.

   $$x=-\frac{-6}{2(1)}=3$$

2. Put 3 back into the function.

   $$f(3)=3^2-6(3)+14$$

3. Work it out.

   $$f(3)=9-18+14=5$$

4. That's the vertex.

   $$(3,5)$$

From factored form, the axis of symmetry is halfway between the zeros. $f(x)=(x-6)(x+12)$ is 0 at 6 and $-12$, so it turns at $\frac{6+(-12)}{2}=-3$. Averaging the printed $-6$ and 12 gives 3, with the sign backward.

Vertex formA quadratic written as a number times (x minus h) squared, plus k, so you can read the vertex (h, k) right off it., $f(x)=a(x-h)^2+k$, shows the vertex $(h,k)$ with no work. The squared part is never negative, so $k$ is the lowest output when $a$ is positive and the highest when $a$ is negative.

With $-3$ in front, no output is above 11, so 11 is the maximum value. A choice calling it the minimum missed the sign of $a$.

Zoom out if the dots are off the screen, or set a taller window with the wrench for a story model with big numbers. A clicked point shows its coordinates as decimals.

Some questions give four equivalent forms and ask which one “displays, as a constant or coefficient,” a feature. All four draw the same graph, so graphing them won't help. Match the feature to the form.

$f(x)=x^2-6x+14$ shows the y-intercept, 14. The same function in vertex form, $f(x)=(x-3)^2+5$, shows the minimum value, 5.

Now try one with a letter in it.

Use the letter for the quantity the question names, write the other quantity with the same letter, then multiply. Say a garden's length $x$, in meters, is 5 meters longer than its width, and its area is 84 square meters.

1. Multiply out $x(x-5)$.

   $$x^2-5x=84$$

2. Subtract 84 from both sides.

   $$x^2-5x-84=0$$

3. $-12$ and 7 multiply to $-84$ and add to $-5$.

   $$(x-12)(x+7)=0$$

4. $x=-7$ also solves it, but a length can't be negative.

   $$x=12$$

A height in feet $t$ seconds after a throw often looks like $h(t)=-16t^2+vt+h_0$. $h_0$ is the starting height, so $h(t)=-16t^2+24t+40$ starts at 40 feet. The object hits the ground when the height is 0, at the positive solution of $-16t^2+24t+40=0$.

An exponential function looks like $f(x)=a\cdot b^x$. $a$ is the starting value, the output at $x=0$. $b$ is the multiplier. Each time $x$ goes up by 1, the output is multiplied by $b$.

Take $P(w)=480(1.15)^w$, the number of members in a club $w$ weeks after it opened sign-ups. At $w=0$, $1.15^0=1$, so only the 480 is left. One week later it's $480(1.15)=552$.

Growing 6% keeps the whole amount and adds 6% of it, so the multiplier is $1+0.06=1.06$. A 20% decrease keeps 80%, so the multiplier is $1-0.20=0.80$.

To go back, subtract 1. A multiplier of 0.85 gives $0.85-1=-0.15$, a 15% decrease, because 85% remains. It's the same multiplier the percentages lesson uses.

The exponent counts how many times the multiplier has been used.

To change the period, put the new period's length in for the input. $S(t)=1{,}500(1.2)^{t/6}$ counts an online group's members $t$ months after it started, so it grows 20% every 6 months. At 12 months the exponent is 2, so each year multiplies the number by $1.2^2=1.44$, a 44% increase.

Percents don't add across periods. The second 20% comes from a bigger amount, so two of them make 44% for the year.

Only the part that changes gets the multiplier. A phone plan with a $15 line fee plus a $40 data charge that rises 5% a year costs $c=15+40(1.05)^t$ after $t$ years. The 15 sits outside the multiplier, so it stays 15.

Here's one with a fixed part and a growth period of 3 weeks.

Two points, given or marked on a graph, pin down an exponential. Say the curve passes through $(0,200)$ and $(2,1800)$.

1. The point at input 0 gives the starting value.

   $$a=200$$

2. Divide the outputs. That's the multiplier across 2 units.

   $$\frac{1800}{200}=9$$

3. Two units means two multipliers, and $3\cdot3=9$.

   $$b=\sqrt{9}=3$$

4. Check: $200\cdot3^2=1800$.

   $$p(x)=200(3)^x$$

To check a choice in Desmos, type it as f(x)=200(3)^x, then f(2) on the next line. It shows 1800, so the curve passes through $(2,1800)$.

In $a\cdot b^{x-c}$, the exponent is 0 at $x=c$. For the value at 0, put 0 in.

1. Put in 0. The exponent is $0-1=-1$.

   $$L(0)=250\left(\frac{5}{8}\right)^{-1}$$

2. A power of $-1$ flips the fraction. $250\div5=50$, and $50\cdot8=400$.

   $$=250\cdot\frac{8}{5}=400$$

Moving the shift can put a value on display, which is how a “displays as a constant or coefficient” question works for an exponential. $M(t)=2048\left(\frac{3}{4}\right)^{t-2}$ for $2\le t\le5$ shrinks the whole way, since its multiplier is less than 1, so its least value is at $t=5$.

1. Put in 5. The exponent is $5-2=3$.

   $$M(5)=2048\left(\frac{3}{4}\right)^{3}$$

2. $2048\div64=32$, and $32\cdot27=864$.

   $$=2048\cdot\frac{27}{64}=864$$

3. Move the shift to 5, and 864 sits in front.

   $$M(t)=864\left(\frac{3}{4}\right)^{t-5}$$

The 2,048 in the original is the value at the other end, $t=2$.

Here's the graph of a function $f$, with its y-intercept marked.

Say $g(x)=f(x)+7$, and the question asks for the y-intercept of $g$. The graph shows $f(0)=-3$.

For a shift inside the parentheses, change the input before you work out $f$. For a shift outside, work out $f$ first and then add or subtract.

Now try one with a graph.

+−100%Reset⤢

The graph of $y=f(x)$ shows Adeyemi Nursery’s watering time during a 10-day cycle.

An adjusted schedule is modeled by $g(x)=f(x)+5$.

What is the $y$-coordinate of the $y$-intercept of the graph of $y=g(x)$?

Show the answer

Accepted answer: 35

Solution

The $y$-intercept occurs at $x=0$. The graph shows $f(0)=30$, and $g(x)=f(x)+5$ shifts every $y$-value up 5 units. Therefore, $g(0)=30+5=35$. A response of $30$ reports the original $y$-intercept without applying the vertical shift. A response of $5$ adds the shift to $0$, the interceptWhere a graph crosses an axis.Read the lesson’s $x$-coordinate, as if $f(0)$ were $0$. The value of $f(0)$ is the intercept’s $y$-coordinate, $30$.

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Before you answer each one, check that you have what the last line asked for.