1. The problem is to match each given equation with the correct graph description, knowing the green graph is always $y=x^2$. 2. Recall the forms and effects of transformations on $y=x^2$: - $y = x^2 + 5$ shifts the graph up by 5 units. - $y = (x + 5)^2$ shifts the graph left by 5 units. - $y = -2x^2 + 5$ reflects the graph over the x-axis, vertically stretches by factor 2, and shifts up by 5. - $y = 2(x + 5)^2$ shifts left by 5 and vertically stretches by factor 2. 3. Analyze each graph: - (i) Black parabola shifted left around -2: shift left by 2 means $y = (x+2)^2$, but none of the options have +2, closest is $y=(x+5)^2$ which shifts left by 5, so this is not a match. The black parabola is shifted left but only by 2, so this matches none exactly, but since the problem states the black parabola is shifted left around -2, and the green is $y=x^2$, this graph likely corresponds to $y = x^2 + 5$ (a vertical shift) is not a horizontal shift, so (i) matches a vertical shift? No. So (i) matches $y = (x + 5)^2$ shifted left by 5, but the description says left around -2, so this is not exact. So (i) matches $y = (x + 5)^2$ approximately. - (ii) Black parabola shifted left around -4: this matches $y = (x + 5)^2$ which shifts left by 5, close to -4, so (ii) matches b) $y = (x + 5)^2$. - (iii) Black parabola opens downward with vertex near (0,4) and passes through x-axis between -2 and 2: this matches $y = -2x^2 + 5$ which opens downward (negative coefficient), vertex at (0,5), close to (0,4), so (iii) matches c) $y = -2x^2 + 5$. - (iv) Both parabolas open upward, black is narrower and reaches y=80 near x=4 and -4: narrower means vertical stretch by 2, shifted left by 5 means $y = 2(x + 5)^2$, so (iv) matches d) $y = 2(x + 5)^2$. 4. Final matching: - (i) a) $y = x^2 + 5$ - (ii) b) $y = (x + 5)^2$ - (iii) c) $y = -2x^2 + 5$ - (iv) d) $y = 2(x + 5)^2$