Question

for numbers 6 to 9, match each of the graphs to the correct equation. write the letter of correct equation: (0.5 marks each)
a.
graph a: parabola opening downward, vertex around (-1, 2), x-intercepts around -3 and 1
b.
graph b: parabola opening downward, vertex at (0, 3), x-intercepts around -2 and 2
c.
graph c: parabola opening downward, vertex at (0.5, 5), x-intercepts around -0.5 and 1.5
d.
graph d: parabola opening upward, vertex at (1, -4), x-intercepts around -1 and 3

1. $y = -0.8x^2 - 0.7x + 3.4$
2. $y = -5.4x^2 + 5.3x + 3.4$
3. $y = 0.8x^2 - 1.6x - 3.1$
4. $y = -0.8x^2 - 1.6x + 1.5$

Explanation:

Step1: Analyze the parabola direction

For a quadratic equation \( y = ax^2+bx + c \), if \( a<0 \), the parabola opens downward; if \( a > 0 \), it opens upward.

• For \( y=- 0.8x^{2}-0.7x + 3.4 \), \( a=-0.8<0 \), opens downward.
• For \( y=-5.4x^{2}+5.3x + 3.4 \), \( a = - 5.4<0 \), opens downward.
• For \( y = 0.8x^{2}-1.6x-3.1 \), \( a=0.8>0 \), opens upward (matches graph d).
• For \( y=-0.8x^{2}-1.6x + 1.5 \), \( a=-0.8<0 \), opens downward.

Step2: Analyze the vertex x - coordinate

The x - coordinate of the vertex of \( y=ax^{2}+bx + c \) is \( x=-\frac{b}{2a} \).

• For \( y=-0.8x^{2}-0.7x + 3.4 \), \( a=-0.8 \), \( b=-0.7 \), \( x =-\frac{-0.7}{2\times(-0.8)}=-\frac{0.7}{1.6}\approx - 0.4375\) (close to x = - 0.5, middle of - 1 and 0).
• For \( y=-5.4x^{2}+5.3x + 3.4 \), \( a=-5.4 \), \( b = 5.3 \), \( x=-\frac{5.3}{2\times(-5.4)}=\frac{5.3}{10.8}\approx0.49\) (close to x = 0.5, middle of 0 and 1, matches graph c).
• For \( y=-0.8x^{2}-1.6x + 1.5 \), \( a=-0.8 \), \( b=-1.6 \), \( x =-\frac{-1.6}{2\times(-0.8)}=-1\) (vertex at x=-1, matches graph a).
• For \( y=-0.8x^{2}-0.7x + 3.4 \), vertex x≈-0.44, and let's check the y - intercept (x = 0, y = 3.4). Graph b has y - intercept around 3 - 4, and vertex x≈-0.5 (close to our calculation). So \( y=-0.8x^{2}-0.7x + 3.4 \) matches graph b.

Step3: Match each equation

• \( y=-0.8x^{2}-0.7x + 3.4 \): opens downward, vertex x≈-0.44, y - intercept≈3.4, matches graph b.
• \( y=-5.4x^{2}+5.3x + 3.4 \): opens downward, vertex x≈0.49, y - intercept≈3.4, matches graph c.
• \( y = 0.8x^{2}-1.6x-3.1 \): opens upward, vertex x=\(-\frac{-1.6}{2\times0.8}=1\), matches graph d (vertex at x = 1, y=\(0.8(1)^2-1.6(1)-3.1=0.8 - 1.6-3.1=-3.9\), close to graph d's vertex).
• \( y=-0.8x^{2}-1.6x + 1.5 \): opens downward, vertex x=-1, y=\(-0.8(1)+1.6 + 1.5=2.3\)? Wait, x=-1: \( y=-0.8(1)-1.6(-1)+1.5=-0.8 + 1.6+1.5 = 2.3\), matches graph a (vertex at x=-1, y≈2 - 3).

Answer:

1. \( y=-0.8x^{2}-0.7x + 3.4 \) matches graph b
2. \( y=-5.4x^{2}+5.3x + 3.4 \) matches graph c
3. \( y = 0.8x^{2}-1.6x-3.1 \) matches graph d
4. \( y=-0.8x^{2}-1.6x + 1.5 \) matches graph a