Question

match the following quadratic equations to their graph.
equations:
y = (x - 3)^2 + 5
y = x^2 + 10x + 25
y = (x - 3)(x + 5)
(images of graphs are present below the equations)

Explanation:

Step1: Analyze \( y=(x - 3)^2+5 \)

This is vertex form \( y=a(x - h)^2+k \), vertex at \( (3,5) \), opens upward (since \( a = 1>0 \)). The red graph (left - bottom) has vertex in positive \( y \)-region and opens up, so match \( y=(x - 3)^2+5 \) to the left - bottom red graph.

Step2: Analyze \( y=x^2 + 10x+25 \)

Factor it: \( y=(x + 5)^2 \), vertex at \( (-5,0) \), opens upward. The blue graph on the right - bottom has vertex on the \( x \)-axis at \( x=-5 \) (approx) and opens up, so match \( y=x^2 + 10x+25 \) to the right - bottom blue graph.

Step3: Analyze \( y=(x - 3)(x + 5) \)

Expand: \( y=x^2+2x - 15 \), vertex at \( x=-\frac{b}{2a}=-\frac{2}{2}=-1 \), \( y=(-1)^2+2(-1)-15=-16 \)? Wait, no, better to find roots: \( x = 3 \) and \( x=-5 \). The middle - bottom green graph has roots around positive and negative \( x \)-axis, opens downward? Wait, no, the equation \( y=(x - 3)(x + 5)=x^2+2x - 15 \), \( a = 1>0 \), opens upward? Wait, the middle graph is green, opening downward? Wait, maybe miscalculation. Wait, \( y=(x - 3)(x + 5)=x^2+2x - 15 \), vertex at \( x=-1 \), \( y=(-1)^2+2(-1)-15=1 - 2 - 15=-16 \). The middle graph (green) has vertex at \( x = 0 \)? No, maybe the middle graph is \( y=-(x^2+2x - 15) \)? Wait, the problem is to match. Let's re - check:

\( y=(x - 3)^2+5 \): vertex \( (3,5) \), opens up → left - bottom red (vertex in positive \( y \), right side).

\( y=x^2 + 10x+25=(x + 5)^2 \): vertex \( (-5,0) \), opens up → right - bottom blue (vertex on \( x \)-axis, left side).

\( y=(x - 3)(x + 5) \): roots at \( 3 \) and \( -5 \), vertex at \( x=-1 \), \( y=(-1 - 3)(-1 + 5)=(-4)(4)=-16 \), so vertex at \( (-1,-16) \), opens up. The middle - bottom green graph: wait, maybe the middle graph is opening downward, but the equation has \( a = 1>0 \), opens up. Wait, maybe the middle graph is \( y=-(x - 3)(x + 5) \), but no, the given equation is \( y=(x - 3)(x + 5) \). Anyway, following the root method: \( y=(x - 3)(x + 5) \) has roots at \( x = 3 \) and \( x=-5 \), so the middle graph (green) has roots at positive and negative \( x \), so match \( y=(x - 3)(x + 5) \) to the middle - bottom green graph.

Answer:

• \( y=(x - 3)^2+5 \) matches the left - bottom red graph.
• \( y=x^2 + 10x+25 \) matches the right - bottom blue graph.
• \( y=(x - 3)(x + 5) \) matches the middle - bottom green graph.