Exercise 13 Page 143

Let's first determine the type of function that best models the height of a plant, and then use the general form of this type to write a function.

Determining the Type of Function

We start by finding the differences of successive y-values, namely the first differences. <jsxgpre id="Solution6477_1" static=1> var b=mlg.board([-2,2.5,11.4,-2],{"desktopSize":"medium","style":"usa"});

var n = 5; /* Number of data values */

var w1=3.5; /* header width */ var w2=1.5; /* cell width */ var hd = 0.1; /* Horizontal spacing */ var v=1; /* Vertical height */ var vd = 0.1*v; /* Vertical spacing */

var i; /* loop counter variable */

/* table background, no need to modify this, it uses the values set above */ b.rectangle([0,0],[w1,2*v],{borders:{visible:false}, fillColor:'#7bcbe8'}); b.rectangle([0,0],[w1+n*w2,2*v],{borders:{strokeWidth:1}}); b.segment([0,v],[w1+n*w2,v],{strokeWidth:1}); for (i = 0; i < n; i++) { b.segment([w1+i*w2,0],[w1+i*w2,2*v],{strokeWidth:1}); }

/* Set the text to write in the header */ b.txt(w1/2-0.75,3*v/2,'Week',{mathMode:false}); b.txt(w1/2,v/2,'Height (in.)',{mathMode:false});

/* Table entries, top row */ b.txt(w1+w2/2,3*v/2,'0'); b.txt(w1+w2/2+w2,3*v/2,'1'); b.txt(w1+w2/2+2*w2,3*v/2,'2'); b.txt(w1+w2/2+3*w2,3*v/2,'3'); b.txt(w1+w2/2+4*w2,3*v/2,'4'); /* Table entries, bottom row */ b.txt(w1+w2/2,v/2,'3'); b.txt(w1+w2/2+w2,v/2,'3.5'); b.txt(w1+w2/2+2*w2,v/2,'4'); b.txt(w1+w2/2+3*w2,v/2,'4.5'); b.txt(w1+w2/2+4*w2,v/2,'5'); /* Add/delete rows as needed.

/* first difference sequence */ b.txt(w1,-v,'First differences:',{mathMode:false, anchorX:'right', strokeColor:'blue'}); for (i = 1; i < n; i++) { b.segment([w1+i*w2-w2/2+hd,-vd],[w1+i*w2-hd,-v/2-vd],{strokeColor:'blue'});

   b.segment([w1+i*w2+w2/2-hd,-vd],[w1+i*w2+hd,-v/2-vd],{strokeColor:'blue'});
   }

b.txt(w1+w2,-v,'0.5',{strokeColor:'blue'}); b.txt(w1+2*w2,-v,'0.5',{strokeColor:'blue'}); b.txt(w1+3*w2,-v,'0.5',{strokeColor:'blue'}); b.txt(w1+4*w2,-v,'0.5',{strokeColor:'blue'}); </jsxgpre> We see that the differences are all equal meaning the table represents a linear function.

Writing the Function

We can model the height y of the plant after x week by using a linear function. y=mx+b Here, m is the slope and b is the y-intercept. From the table, we see that the y-intercept is 3. Let's put this into the equation using substitution. y=mx+b ⇕ y=mx+3 We can find the slope m by substituting an ordered pair in the table. Let's substitute (2,4).

The slope of the linear function is 12=0.5. Then, the function below models the data. \begin{gathered} y=\textcolor{darkorange}{m}x+3 \\ \Updownarrow \\ y=\textcolor{darkorange}{0.5}x+3 \end{g

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