Quadratics

4.2 A Steepness Factor

Lesson materials located below the video overview.

Change 4: Steepness

Here is the graph \(y=x^{2}\) yet again:

Q4_2_pic1

Q4_2_pic2

What can we say about the graph of \(y=2x^{2}\) ? Certainly all the outputs are doubled:

Q4_2_pic3

This creates a “steeper” U-shaped graph:

Q4_2_pic4

 

YOUR TURN…

Recall that all solutions to the proactice problems appear in the COMPANION GUIDE to this QUADRATICS course.

 

PRACTICE 61:

a) Draw a table of values for \(y=3x^{2}\) and sketch its graph.

b) Which graph is “steeper,” that for \(y=100x^{2}\) or that for \(y=200x^{2}\)?

 

Consider \(y=-x^{2}\). It’s outputs are the negative of the outputs for \(y=x^{2}\) :

Q4_2_pic5

Q4_2_pic6

 

PRACTICE 62: Describe the graph of \(y=-2x^{2}\).

 

PRACTICE 63:

a) Quentin says that the graph of \(y= \frac{1}{10} x^{2}\) will be a very “broad” upward facing U-shaped graph. Is he right? Explain.

b) Describe the graph of  \(y=-\frac{1}{1000} x^{2}\).

PUTTING IT ALL TOGETHER

 

EXAMPLE 64: Analyse and quickly sketch \(y=2(x-3)^{2}+1\) .

 

Answer: Now \(y=2(x-3)^{2}+1\) is essentially the graph \(y=x^{2}\) “messed around with.”

We see it is a steep U-shaped graph (“steepness \(2\)”) with \(x=3\) behaving like zero for the \(x\)-values, and the whole graph is shifted one unit high.

Q4_2_pic7

 

EXAMPLE 65: Analyse and quickly sketch \(y=-(x+2)^{2}+4\).

 

Answer: \(y=-(x+2)^{2}+4\) is  \(y=x^{2}\)  messed around with. It is an upside-down U with \(x=-2\) made the new zero and shifted upwards four units.

Q4_2_pic8

 

EXAMPLE 66: Write a formula for this symmetric U-shaped graph:

Q4_2_pic9

Answer: We see that is a U-shaped graph with \(x=4\) as its new zero. There is no vertical shifting. It is not clear what the steepness would be. We can write:

\(y=a(x-4)^{2}\)

and see if we can determine the steepness \(a\).

The graph passes through the point \(x=0\), \(y=6\). Let’s substitute in those values:

\(6=a(0-4)^{2}\)

\(6=16a\)

\(a=\frac{6}{16} = \frac{3}{8}\)

Thus

\(y=\frac{3}{8} (x-4)^{2}\).  □

 

EXAMPLE 67: Write a formula for this symmetric U-shaped graph:

Q4_2_pic10

Answer: This is the graph \(y=x^{2}\) messed around with.

Because it is symmetric, we see that it has its peak at \(x=6\). So \(x=6\) is behaving like zero for the \(x\)-values. Also, the graph is shifted upwards \(40\) places.

We don’t know the steepness – though it should be negative. We can at least write:

\(y=a(x-6)^{2}+40\)

but we need another piece of information to determine \(a\) .

 

We could try plugging in \(x=6\), \(y=40\) but it doesn’t help:

\(40=a(0)^{2}+40\)

\(40=40\)

Actually it is not surprising that it fails to help: we have already made use of the values \(x=6\) and \(y=40\) to get this far. Let’s try plugging in value of a point we haven’t yet used.

 

The graph passes through \(x=0\), \(y=0\). Let’s try this:

\(0=a(-6)^{2}+40\)

\(-40=36a\)

\(a=-\frac{36}{40} = -\frac{9}{10}\)

Bingo!  And we have negative steepness as expected.

 

NOTE: We could have also tried \(x=12\), \(y=0\). It also gives \(a=-\frac{9}{10}\) . (Try it!)

 

So we now know the formula for this U-shaped graph:  \(y=-\frac{9}{10} (x-6)^{2}+40\)   □

 

EXAMPLE 68: Write down the formula of a U-shaped graph that passes through the points \( \left( {3,18} \right) \) and \(\left( {17,18} \right) \) and has lowest value \(5\).

 

Answer:

PICTURES SPEAK A THOUSAND WORDS!
It is always good to draw a sketch.

This graph must look something like:

Q4_2_pic11

Because it is symmetrical, we see that the point halfway between \(3\) and \(17\), namely \(x=10\), is the new zero for the \(x\)-values. The whole graph is shifted up \(5\) units. We have:

\(y=a(x-10)^{2}+5\)

Putting in \(x=3\), \(y=18\) gives:

\(18=49a+5\)

\(a=\frac{13}{49}\)

So

\(y=\frac{13}{49}(x-6)^{2}+5\).  □

 

Here is a hard question that will be easier for us once when we learn one more thing about these U-shaped graphs later on. But it is good to see that even these questions are within our reach at this stage.

 

EXAMPLE 69: Write down the formula of a U-shaped graph that passes through the \(x\)-axis at \(x=-2\) and at \(x=10\) and the \(y\)-axis at \(y=-6\).

 

Answer: Again let’s make a sketch.

Q4_2_pic12

Because the graph is symmetrical we see that \(x=4\) is the new zero for the \(x\)-values.

We don’t know the steepness and we don’t know how far the graph is shifted down.

The best we can write right now is:

\(y=a(x-4)^{2}+b\)

 

Let’s plug in some points we know:

\(x=-2\), \(y=0\) gives:  \(0=36a+b\)

\(x=0\),\(y=-6\) gives: \(-6=16a+b\)

 

We now have a system of two equations in two unknowns:

\(0=36a+b\)

\(-6=16a+b\)

 

Subtracting gives:

\(6=20a+0\)

and so

\(a=\frac{6}{20}=\frac{3}{10}\)   (and we expected positive steepness).

This means:

\(b=-36a=-\frac{36 \cdot 3}{10}=-\frac{18 \cdot 3}{5} = -\frac{54}{5}\).

 

Thus our equation is:

\(y=\frac{3}{10}(x-4)^{2}-\frac{54}{5}\).   □

 

PRACTICE 70: Make reasonable sketches of the following graphs:

a) \(y=3(x-5)^{2}-3\)

b) \(y=2-x^{2}\)

c) \(y=\frac{1}{3}(x-2)^{2}-4\)

d) \(y=0.034(x+0.276)^{2}+0.778\)

e) \(y=2000000(x-2000000)^{2}-2000000\)

 

PRACTICE 71: Write formulas for each of the following symmetric U-shaped graphs:

Q4_2_pic13

Q4_2_pic14

Q4_2_pic15

 

PRACTICE 72:

a) Write down the formula for a symmetric U-shaped graph that crosses the \(x\)-axis at \(x=-5\) and \(x=17\), and has highest value \(8\).

b) Write down the formula for a symmetric U-shaped graph that passes through the points \( \left( {4},{10} \right) \) and \( \left( {6},{10} \right) \) and \( \left( {8},{13} \right) \).

c) Write down the formula for a symmetric U-shaped graph that has \(x\)-intercepts \(6\) and \(8\) and \(y\)-intercept \(-12\).

 

SOME LANGUAGE:

The place where a U-shaped graph dips down to its lowest point, if it is upward facing, or “dips up” to its highest point if it is downward facing is called the vertex of the quadratic graph.

Q4_2_pic16

COMMENT: Some teachers might be surprised I am resisting calling these curves parabolas. Part 6 of this course explains my hesitation.

 

PRACTICE 73: Find the formula for the quadratic U-curve with vertex \( \left( {-3},{7} \right) \) that passes through \( \left( {10},{10} \right) \). [HINT: As usual, sketch a picture first.]

 

Self Check: Question 11

Another round of questions to check our understanding!


Self Check: Question 12

A second question!

Self Check: Question 13

A third question of joy!

Self Check: Question 14

A fourth question:

Self Check: Question 15

A fifth and final question.

 

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