## Math 10 Chapter 2 Lesson 3: Quadratic Functions

## 1. Summary of theory

### 1.1. Define

### 1.2. Graph of quadratic function

\(\begin{array}{l} a{x^2} + bx + c = a\left( {{x^2} + 2\frac{b}{{2x}} + \frac{{{b) ^2}}}{{4{a^2}}}} \right) – \frac{{{b^2}}}{{4{a^2}}} + c\\ = a{\left ( {x + \frac{b}{{2a}}} \right)^2} – \frac{{{b^2} – 4ac}}{{4a}} \end{array}\)

So if set: \(\Delta = {b^2} – 4ac;p = – \frac{b}{{2a}};q = – \frac{\Delta }{{4a}}\)

Then the function \(y=ax^2+bx+c(a\ne0)\) become \(y = a{\left( {x – p} \right)^2} + q\)

Graph of function \(y=ax^2+bx+c(a\ne0)\) is a parabola with vertices \(I\left( { – \frac{b}{{2a}}; – \frac{\Delta }{{4a}}} \right)\), get straight line \(x = – \frac{b}{{2a}}\) make the axis of symmetry and point the concave upward when a is positive, and the concave downward when a is negative.

### 1.3. Variation of the quadratic function

## 2. Illustrated exercise

**Question 1: **Define the parabola \(\left( P \right)\): \(y = a{x^2} + bx + c\), \(a \ne 0\) knows \(\left( P \right) \) passes through \(M(4; 6)\) which has vertex \(I(2; 4)\).

**Solution guide**

Since \(M \in \left( P \right)\) \(6 = 16a + 4b + c\) (1).

Otherwise \(\left( P \right)\) has vertex \(I(2;4)\) so \( – \frac{b}{{2a}} = 2 \Leftrightarrow 4a + b = 0\) (2) and \(I \in \left( P \right)\) deduce \(4 = 4a + 2b + c\) (3)

From (1), (2) and (3) we have \(\left\{ \begin{array}{l}6 = 16a + 4b + c\\4a + b = 0\\4 = 4a + 2b + c\end{array} \right.\Leftrightarrow \left\{ \begin{array}{l}a = 1/2\\b = – 2\\c = 6\end{array} \right.\)

So \(\left( P \right)\) to find is \(y = 1/2{x^2} – 2x + 6\).

**Verse 2: **Determine the parabola \(\left( P \right)\): \(y = a{x^2} + bx + c\), \(a \ne 0\) knows the Function \(y = a{x) ^2} + bx + c\) has a minimum value of \(2\) when \(x = 1\) and is equal to \(1\) when \(x = 6\).

**Solution guide**

The function \(y = a{x^2} + bx + c\) has the smallest value equal to \(2\) when \(x = 1\) so we have:

\(- \frac{b}{{2a}} = 1 \Leftrightarrow 2a + b = 0\) (5)

\(2 = a{.1^2} + b.1 + c \Leftrightarrow a + b + c = 2\) (6)

The function \(y = a{x^2} + bx + c\) takes the value \(6\) when\(x = -1\) so \(a -b + c = 6\)(7 )

From (5), (6) and (7) we have \(\left\{ \begin{array}{l}2a + b = 0\\a + b + c = 2\\a – b + c = 6\end{array} \right \Leftrightarrow \left\{ \begin{array}{l}a = 1\\b = – 2\\c = 3\end{array} \right.\)

So \(\left( P \right)\) to find is \(y = {x^2} -2 x + 3\).

**Question 3: **Make a table of variables and graph the function \(y = – {x^2} + 2\sqrt 2 x\)

**Solution guide:**

We have \( – \frac{b}{{2a}} = \sqrt 2 ,\,\, – \frac{\Delta }{{4a}} = 2\)

Variation table:

Infer the graph of the function \(y = – {x^2} + 2\sqrt 2 x\) whose vertex is \(I\left( {\sqrt 2 ;2} \right)\), passing through the points \(O\left( {0;0} \right),\,\,B\left( {2\sqrt 2 ;0} \right)\)

Take the line \(x = \sqrt 2 \) as the axis of symmetry and point the concave downward.

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Define the parabola \(\left( P \right)\): \(y = a{x^2} + bx + c\), \(a \ne 0\) knows \(\left( P \right) \) passes through \(A(-2; 4)\) with vertex \(I(2;-2)\).

**Verse 2: **Determine the parabola \(\left( P \right)\): \(y = a{x^2} + bx + c\), \(a \ne 0\) knows the Function \(y = a{x) ^2} + bx + c\) has a minimum value of \(\frac{2}{3}\) when \(x = \frac{1}{3}\) and takes the value \(3 \) when\(x = 1\).

**Question 3: **Make a table of variations and plot the following functions:

a) \(y = {x^2} + 5x + 6\)

b) \(y = {x^2} + 3\sqrt 3 x\)

### 3.2. Multiple choice exercises

**Question 1: **Which of the following functions is inverse in the range \(\left( { – \infty ;\,0} \right)\)?

A. \(y = \sqrt 2 {x^2} + 1\)

B. \(y = – \sqrt 2 {x^2} + 1\)

C. \(y = \sqrt 2 {\left( {x + 1} \right)^2}\)

D. \(y = – \sqrt 2 {\left( {x + 1} \right)^2}\)

**Verse 2: **Given the function: \(y = {x^2} – 2x + 3\). Which of the following statements is correct?

A. y increments on \(\left( {0;\, + \infty } \right)\)

B. y decreases on \(\left( { – \infty ;\,2} \right)\)

C. The graph of y has vertices I(1,0)

D. y increments on \(\left( {2;\, + \infty } \right)\)

**Question 3: **The coordinates of the vertex I of the parabola \(\left( P \right):y = 2{x^2} – 4x + 3\) are:

A. -1

B. 1

C. 5

D. -5

**Question 4: **Which of the following functions has the smallest value at \(x = \frac{3}{4}\)?

A. \(y = 4{x^2}-3x{\rm{ }} + 1\)

B. \(y = – {x^2} + \frac{3}{2}x + 1\)

C. \(y = -2{x^2} + 3x + 1\)

D. \(y = {x^2} – \frac{3}{2}x + 1\)

** Question 5: **Let the function \(y = f\left( x \right) = – {x^2} + 4x + 2\). Which of the following statements is true?

A. y decreases on \(\left( {2;\, + \infty } \right)\)

B. y decreases on \(\left( { – \infty ;\,2} \right)\)

C. y increments on \(\left( {2;\, + \infty } \right)\)

D. y increments on \(\left( { – \infty ;\,2} \right)\)

## 4. Conclusion

Through this lesson, you should be able to understand the following:

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