## Math 9 Chapter 4 Lesson 1: Function \(y = ax^2\) (a 0)

## 1. Summary of theory

### 1.1. Properties of the function \(y=ax^2\) (a≠0)

**a) Deterministic set of the function \(y = a{x^2}\) \((a ≠ 0)\)**

The function \(y = a{x^2}\) \((a ≠ 0)\) determines for all values \(x ∈ R.\) so the set \(D=R.\)

**b) Properties**

Consider the function \(y = a{x^2}\) \((a ≠ 0)\)

– If \(a > 0\), the function is inverse when \(x < 0\) and covariate when \(x > 0\).

– If \(a < 0\), the function is covariant when \(x < 0\) and inverse when \(x > 0\).

### 1.2. Comment

Consider the function \(y = a{x^2}\) \((a ≠ 0)\)

– If \(a > 0\) then \(y > 0\) for all \(x ≠ 0; y = 0\) when \(x = 0\). Minimum value of the function \(y = 0\).

– If \(a < 0\) then \(y < 0\) for all \(x ≠ 0; y = 0\) when \(x = 0\). The maximum value of the function is \(y = 0\).

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1: **For the function \(y = 2{x^2}\), thanks to the tables of values just calculated, please tell us

– When \(x\) increases but is always negative, does the corresponding value of \(y\) increase or decrease?

– When \(x\) increases but is always positive, does the corresponding value of \(y\) increase or decrease?

Similar comment with function \(y = – 2{x^2}\)

**Solution guide**

* From the table of values of the function \(y = 2{x^2}\) we see

– When \(x\) increases but is always negative, the corresponding value of \(y\) decreases.

– When \(x\) increases but is always positive, the corresponding value of \(y\) increases.

* From the table of values of the function \(y = – 2{x^2}\) we see

– When \(x\) increases but is always negative, the corresponding value of \(y\) increases.

– When \(x\) increases but is always positive, the corresponding value of \(y\) decreases.

**Verse 2:** Given the function \(y=-2x^2\). Calculate the value of \(y\) with \(x=3; x=-2; x=6\)

**Solution guide**

With \(x=3\Rightarrow y=-2.3^2=-18\)

Similarly \(x=3\Rightarrow y=-2.(-2)^2=-8\)

\(x=3\Rightarrow y=-2.6^2=-72\)

**Comment:** The sign of the function y depends on the sign of the coefficient a!

**Question 3:** In a circle:

If the radius is increased by 3 times, how many times does the area increase?

If the area is reduced by 16 times, how will the radius change?

**Solution guide**

We have the formula to calculate the area of a circle: \(S=\pi R^2\)(Where R is the radius of the circle)

So if the radius is increased 3 times, then \(S’=\pi R’^2=\pi (3R)^2=9S\), so the area increases 9 times.

Similarly, if the area is reduced by 16 times, the radius will decrease by \(\sqrt{16}=4\) (times)

### 2.2. Advanced exercises

**Question 1:** Find the radius of a circle with an area equal to \(16\pi ^2\) (cm)

**Solution guide**

The area of a circle has the formula \(S=\pi R^2\Leftrightarrow 16\pi ^2=\pi R^2\Rightarrow R=4\sqrt{\pi} (cm)\)

**Verse 2:** An object falls freely from a height of 400 m above the ground. The distance traveled by the falling body depends on time by the formula \(s=4t^2\)( distance s (m), time t(s)). So how long does it take for this object to hit the ground?

**Solution guide**

\(s=4t^2\) \(\Leftrightarrow 400=4t^2\Rightarrow t=10s\)

## 3. Practice

### 3.1. Essay exercises

**Question 1:** Given the function \(y = 3{x^2}\)

a) Make a spreadsheet of the values of \(y\) corresponding to the values of \(x\) by: \( – 2; – 1; – \displaystyle {1 \over 3};0;{ 1 \over 3};1;2\)

b) On the coordinate plane determine the points where the coordinate is the value of \(x\) and the coordinate is the corresponding value of y found in the sentence \(a,\) (for example, the point \( A\left( { -\displaystyle {1 \over 3};{1 \over 3}} \right)\)

**Verse 2:** Given the function \(y = – 3{x^2}.\)

a) Make a spreadsheet of the values of \(y\) corresponding to the values of \(x\) by: \( – 2; – 1; -\displaystyle {1 \over 3};0;{ 1 \over 3};1;2\)

b) On the coordinate plane determine the points where the coordinate is the value of \(x\) and the coordinate is the corresponding value of \(y\) found in the sentence \(a,\) (eg. , point \(A\left( \displaystyle{ – {1 \over 3};-{1 \over 3}} \right)\))

**Question 3: **Given the function \(y = f\left( x \right) = – 1.5{x^2}\)

a) Calculate \(f\left( 1 \right),f\left( 2 \right),f\left( 3 \right)\) and then arrange these three values in order from largest to smallest.

b) Calculate \(f\left( { – 3} \right),f\left( { – 2} \right),f\left( { – 1} \right)\) and then arrange these three numbers in order from baby to adult.

c) State your comments on the covariance or inverse of this function when \(x > 0;\) when \(x < 0.\)

**Question 4:** An object falls from a height of 100 m above the ground. The distance traveled s (meters) of the falling body depends on the time t (seconds) by the formula \(\small s = 4t^2\)

a) After 1 second, how far is the object from the ground? Similarly, after 2 seconds?

b) How long will it take for the object to hit the ground?

### 3.2. Multiple choice exercises

**Question 1:** If the area of a circle is increased by 4 times, then its radius will be:

A. 2 times reduction

B. increase 2 times

C. reduced by 4 times

D. increase 4 times

**Verse 2: **The value of the function \(y= -4x^2\) is \(-44\) at the point x equals:

A. \(\sqrt{11}\)

B. \(-\sqrt{11}\)

C. \(11\)

D. \(\pm \sqrt{11}\)

**Question 3:** The value of the function \(y=0.1x^2\) at \(x=6\) is:

A. \(360\)

B. \(0.36\)

C. \(36\)

D. \(3,6\)

**Question 4:** Given the function \(y=ax^2 (a>0)\). So the value of \(y\):

A. always non-negative

B. always negative

C. is equal to the constant

D. depends on the variable x

**Question 5: **A circle and a rectangle have equal areas. If the length is increased by 4 times and the width of the rectangle is decreased by 9 times, we get a rectangle with a new area. So, for the original circle to have an area equal to the new rectangle, how must the radius of the circle change?

A. 1.5 times increase

B. 1.5 times reduction

C. increase 2 times

D. decrease 2 times

## 4. Conclusion

This lesson helps students:

- Know how to calculate the value of a function that corresponds to a given value of a variable.
- Calculate the value of the function corresponding to the given value of the variable.

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