Illustrative Examples: How 'a' Changes Quadratic Graph Shape

📚 Quick Study Guide

🔍 General Form: A quadratic equation is generally represented as $f(x) = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants.

📈 Effect of 'a':
• 📏 If $a > 0$, the parabola opens upwards. The graph has a minimum point.
• 📉 If $a < 0$, the parabola opens downwards. The graph has a maximum point.
• ↔️ The larger the absolute value of 'a' ($|a|$), the narrower (steeper) the parabola.
• ↕️ The smaller the absolute value of 'a' ($|a|$), the wider (flatter) the parabola.

Practice Quiz

1. Which of the following statements is true about the graph of $y = ax^2 + bx + c$ when $a < 0$?
1. The parabola opens upwards.
2. The parabola opens downwards.
3. The parabola is narrower than $y = x^2$.
4. The parabola has a minimum point.
2. How does increasing the absolute value of 'a' in the quadratic equation $y = ax^2$ affect the graph?
1. It makes the parabola wider.
2. It makes the parabola flatter.
3. It makes the parabola narrower.
4. It shifts the parabola vertically.
3. What happens to the parabola $y = x^2$ if 'a' is changed to -1, resulting in $y = -x^2$?
1. The parabola shifts to the right.
2. The parabola shifts upwards.
3. The parabola is reflected across the x-axis.
4. The parabola becomes wider.
4. Which of the following quadratic equations has the widest parabola?
1. $y = 3x^2$
2. $y = 0.5x^2$
3. $y = -2x^2$
4. $y = -4x^2$
5. If the graph of $y = ax^2 + bx + c$ has a minimum value, what can you conclude about the value of 'a'?
1. $a < 0$
2. $a = 0$
3. $a > 0$
4. $a = 1$
6. How does changing 'a' from 1 to 4 in the equation $y = ax^2$ change the graph?
1. The parabola becomes wider.
2. The parabola opens downwards.
3. The parabola becomes narrower.
4. The parabola shifts upwards.
7. Which equation represents a parabola that opens downward and is narrower than $y = -x^2$?
1. $y = -0.5x^2$
2. $y = -2x^2$
3. $y = 0.5x^2$
4. $y = 2x^2$