To solve the quadratic equation:

 

$$x^2 – 5x + 6 = 0$$ 

Step 1: Factor the quadratic

We look for two numbers that multiply to 6 and add up to -5.

Those numbers are -2 and -3.

 

$$x^2 – 5x + 6 = (x – 2)(x – 3)$$ 

Step 2: Set each factor to zero

 

$$x – 2 = 0 \quad \text{or} \quad x – 3 = 0$$ 

 

Step 3: Solve for $x$

 

 

 

$$x = 2 \quad \text{or} \quad x = 3$$

 

 

 

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Final Answer:

$$x = 2 \quad \text{or} \quad x = 3$$

 

 

 

Would you like this in a formatted table or webpage layout as well?

More Examples on the Order of Transformations of Functions

Example 1: Transform $f(x) = x^2$

 

 

 

 

into $g(x) = -3(x – 2)^2 + 4$

 

 

 

 

We analyze the transformations step by step:

1. Horizontal Translation:
   $x – 2$ 

   → Shift right by 2.

2. Vertical Stretch:
   $3f(x)$ 

   → Stretch by factor of 3.

3. Reflection over the x-axis:
   $-3 f(x)$ 

   → Flip upside down.

4. Vertical Translation:
   $+4$ 

   → Shift up by 4.

Graphing Process:

• Start with
  $f(x) = x^2$ 

  .

• Shift right by 2.
• Stretch vertically by 3 (parabola becomes narrower).
• Reflect over the x-axis (opens downward).
• Shift up by 4.

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Example 2: Transform $f(x) = |x|$

 

 

 

 

into $g(x) = 2|x + 1| – 3$

 

 

 

 

Step-by-step transformations:

1. Horizontal Translation:
   $x + 1$ 

   → Shift left by 1.

2. Vertical Stretch:
   $2f(x)$ 

   → Stretch vertically by a factor of 2 (steeper V-shape).

3. Vertical Translation:
   $-3$ 

   → Shift down by 3.

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Example 3: Transform $f(x) = \sqrt{x}$

 

 

 

 

into $g(x) = -\frac{1}{2} \sqrt{2(x – 3)} + 5$

 

 

 

 

Step-by-step transformations:

1. Horizontal Stretch/Compression:
   $2(x – 3)$ 

   → Horizontal compression by factor of

   $\frac{1}{2}$ 

   .

2. Horizontal Translation:
   $x – 3$ 

   → Shift right by 3.

3. Vertical Stretch/Compression:
   $\frac{1}{2} f(x)$ 

   → Vertical compression by factor of

   $\frac{1}{2}$ 

   .

4. Reflection over x-axis:
   $-\frac{1}{2} f(x)$ 

   → Flip upside down.

5. Vertical Translation:
   $+5$ 

   → Shift up by 5.

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Example 4: Transform $f(x) = \sin x$

 

 

 

 

into $g(x) = -4\sin(0.5(x – \pi)) + 2$

 

 

 

 

Step-by-step transformations:

1. Horizontal Stretch:
   $0.5(x – \pi)$ 

   → Stretch horizontally by a factor of 2.

2. Horizontal Translation:
   $x – \pi$ 

   → Shift right by

   $\pi$ 

   .

3. Vertical Stretch:
   $4f(x)$ 

   → Stretch vertically by a factor of 4.

4. Reflection over the x-axis:
   $-4 f(x)$ 

   → Flip upside down.

5. Vertical Translation:
   $+2$ 

   → Shift up by 2.

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Example 5: Transform $f(x) = e^x$

 

 

 

 

into $g(x) = -e^{3(x+2)} – 7$

 

 

 

 

Step-by-step transformations:

1. Horizontal Stretch/Compression:
   $3(x + 2)$ 

   → Compress horizontally by a factor of

   $\frac{1}{3}$ 

   .

2. Horizontal Translation:
   $x + 2$ 

   → Shift left by 2.

3. Reflection over the x-axis:
   $-f(x)$ 

   → Flip upside down.

4. Vertical Translation:
   $-7$ 

   → Shift down by 7.

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Example 6: Transform $f(x) = \frac{1}{x}$

 

 

 

 

into $g(x) = \frac{-2}{(x – 5)} + 4$

 

 

 

 

Step-by-step transformations:

1. Horizontal Translation:
   $x – 5$ 

   → Shift right by 5.

2. Vertical Stretch:
   $2f(x)$ 

   → Stretch vertically by a factor of 2.

3. Reflection over the x-axis:
   $-2 f(x)$ 

   → Flip upside down.

4. Vertical Translation:
   $+4$ 

   → Shift up by 4.

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Key Takeaways from These Examples

1. Horizontal transformations (inside
   $f(x)$ 

   ) are applied first.

2. Vertical transformations (outside
   $f(x)$ 

   ) are applied last.

3. Reflections must be done before translations.
4. Always sketch step-by-step to see how the function changes.

Would you like some GeoGebra or Desmos-based activities for visualization? 😊

solve the quadratic equation:

 

$$x^2 – 5x + 5 = 0$$ 

We use the quadratic formula:

 

$$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$ 

Here,

$a = 1$,

$b = -5$,

$c = 5$ 

 

$$x = \frac{-(-5) \pm \sqrt{(-5)^2 – 4(1)(5)}}{2(1)} = \frac{5 \pm \sqrt{25 – 20}}{2} = \frac{5 \pm \sqrt{5}}{2}$$ 

Final Answer:

 

$$x = \frac{5 \pm \sqrt{5}}{2}$$’

$$x^2 – 5x + 5 = 0$$We use the quadratic formula:

 

$$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$Here,

$a = 1$,

$b = -5$,

$c = 5$ 

 

$$x = \frac{-(-5) \pm \sqrt{(-5)^2 – 4(1)(5)}}{2(1)} = \frac{5 \pm \sqrt{25 – 20}}{2} = \frac{5 \pm \sqrt{5}}{2}$$ 

Final Answer:

 

$$x=\frac{5\pm \sqrt{5}}{2}$$ 

$$ $$$$

IntroductionHere’s the rewritten question and solution for 

 

$f(x) = -3x^2 + 6x – 2$, with equations and calculations inline:

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For the quadratic function

$f(x) = -3x^2 + 6x – 2$:

(a) Express the quadratic function in its standard form by completing the square.

(b) Find:

• The vertex of the quadratic function.
• The x-intercepts.
• The y-intercept.

(c) Write the equation of the axis of symmetry.

(d) Sketch the graph of the quadratic function.

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Solution:

(a) Standard Form:

1. Factor out the coefficient of
   $x^2$ 

   from the first two terms:

   $$f(x) = -3(x^2 – 2x) – 2$$ 

2. Complete the square for
   $x^2 – 2x$ 

   :
   Add and subtract

   $\left(\frac{-2}{2}\right)^2 = 1$ 

   :

   $$f(x) = -3\left(x^2 – 2x + 1 – 1\right) – 2$$Simplify:

    

   $$f(x) = -3\left((x – 1)^2 – 1\right) – 2$$ 

3. Distribute the
   $-3$ 

   and combine constants:

   $$f(x) = -3(x – 1)^2 + 3 – 2$$ 

   $$f(x) = -3(x – 1)^2 + 1$$ 

Vertex Form:

 

$$f(x) = -3(x – 1)^2 + 1$$ 

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(b) Find:

1. Vertex:
   The vertex is at $(h, k) = (1, 1)$ 

   .

2. X-intercepts:
   Solve $f(x) = 0$ 

   :

   $$-3(x – 1)^2 + 1 = 0$$Simplify:

    

   $$-3(x – 1)^2 = -1$$Divide through by

   $-3$:

    

   $$(x – 1)^2 = \frac{1}{3}$$Take the square root:

    

   $$x – 1 = \pm \sqrt{\frac{1}{3}}$$Solve for

   $x$:

    

   $$x = 1 \pm \frac{\sqrt{3}}{3}$$X-intercepts:

   $\left(1 + \frac{\sqrt{3}}{3}, 0\right)$and

   $\left(1 – \frac{\sqrt{3}}{3}, 0\right)$.

3. Y-intercept:
   Substitute $x = 0$ 

   :

   $$f(0) = -3(0 – 1)^2 + 1$$Simplify:

    

   $$f(0) = -3(1) + 1 = -3 + 1 = -2$$Y-intercept:

   $(0, -2)$.

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(c) Equation of the Axis of Symmetry:
The axis of symmetry is the vertical line passing through the vertex:

 

$$x = 1$$ 

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(d) Sketch:
The graph is a parabola opening downwards, with the following key features:

• Vertex:
  $(1, 1)$ 
• X-intercepts:
  $\left(1 + \frac{\sqrt{3}}{3}, 0\right)$ 

  and

  $\left(1 – \frac{\sqrt{3}}{3}, 0\right)$ 

• Y-intercept:
  $(0, -2)$ 
• Axis of symmetry:
  $x = 1$ 

Would you like me to generate the graph for this function?

To sol

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St

$$x = 2 \quad \text{and} \quad x = 3$$

Exa units.

Question

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