## 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:

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

This creates a “steeper” U-shaped graph:

**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 \(y=x^{2}\) :

**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.

**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.

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

**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*:

**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:

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

\(y=\frac{13}{49}(x-10)^{2}+5\). □

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.

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:*

**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.

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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