Question

p(x) = 4(1.5)^x + 3; m(x) = 4(1.5)^x - 6; t(x) = 1.5^x + 3; w(x) = 1.5^x - 6

Explanation:

Step1: Analyze Vertical Stretch/Compression

For exponential functions of the form \( y = a(b)^x + k \), the coefficient \( a \) affects vertical stretch/compression. If \( |a|>1 \), it's a vertical stretch; if \( 0<|a|<1 \), it's a vertical compression. Here, \( p(x) \) and \( m(x) \) have \( a = 4 \) (stretch), \( t(x) \) and \( w(x) \) have \( a = 1 \) (no stretch/compression). So top - left and bottom - left graphs (with steeper curves) correspond to \( p(x) \) or \( m(x) \); top - right and bottom - right (less steep) to \( t(x) \) or \( w(x) \).

Step2: Analyze Vertical Shift (\( k \))

The vertical shift \( k \) is the horizontal asymptote (HA) of the exponential function \( y=a(b)^x + k \), so \( HA:y = k \).
- For \( p(x)=4(1.5)^x + 3 \), \( HA:y = 3 \) (shift up 3).
- For \( m(x)=4(1.5)^x-6 \), \( HA:y=- 6 \) (shift down 6).
- For \( t(x)=1.5^x + 3 \), \( HA:y = 3 \) (shift up 3).
- For \( w(x)=1.5^x-6 \), \( HA:y=-6 \) (shift down 6).

Step3: Match Graphs to Functions

- Top - left graph: Steep ( \( a = 4 \) ), HA around \( y=-6 \)? Wait, no. Wait, let's check the y - intercept. For \( x = 0 \):
- \( p(0)=4(1)+3 = 7 \)
- \( m(0)=4(1)-6=-2 \)
- \( t(0)=1 + 3 = 4 \)
- \( w(0)=1-6=-5 \)
Top - left graph: At \( x = 0 \), \( y\approx - 2 \)? Wait, no, the top - left graph at \( x = 0 \) is around \( y=-2 \)? Wait, let's re - check.
Wait, first, steepness: \( p(x) \) and \( m(x) \) have \( a = 4 \), so they are steeper than \( t(x) \) and \( w(x) \) ( \( a = 1 \) ).
- Bottom - left graph: Let's check \( x = 0 \). For \( m(x)=4(1.5)^0-6=4 - 6=-2 \)? No, wait \( m(0)=4 - 6=-2 \), \( p(0)=4 + 3 = 7 \). The bottom - left graph at \( x = 0 \) is \( y\approx - 6 \)? Wait, no, let's look at the horizontal asymptote.
For \( p(x)=4(1.5)^x + 3 \), as \( x ightarrow-\infty \), \( 4(1.5)^x ightarrow0 \), so \( y ightarrow3 \). So the graph should approach \( y = 3 \) as \( x ightarrow-\infty \).
For \( m(x)=4(1.5)^x-6 \), as \( x ightarrow-\infty \), \( y ightarrow - 6 \).
For \( t(x)=1.5^x + 3 \), as \( x ightarrow-\infty \), \( y ightarrow3 \).
For \( w(x)=1.5^x-6 \), as \( x ightarrow-\infty \), \( y ightarrow - 6 \).

Now, let's match:
- Top - left graph: Steep ( \( a = 4 \) ), approaches \( y=-6 \) as \( x ightarrow-\infty \) (so \( m(x) \))? Wait, no, when \( x = 0 \), \( m(0)=-2 \), and the top - left graph at \( x = 0 \) is around \( y=-2 \)? Wait, the top - left graph has a y - intercept around \( y=-2 \), and is steep ( \( a = 4 \) ), so \( m(x)=4(1.5)^x-6 \) (since \( m(0)=-2 \), steep, HA \( y=-6 \)).
- Bottom - left graph: Steep ( \( a = 4 \) ), approaches \( y = 3 \) as \( x ightarrow-\infty \) (HA \( y = 3 \)), and at \( x = 0 \), \( p(0)=7 \). The bottom - left graph at \( x = 0 \) is around \( y=-6 \)? No, wait, I think I messed up. Wait, let's take the bottom - left graph: when \( x = 4 \), let's see. For \( m(x)=4(1.5)^4-6=4\times5.0625-6 = 20.25 - 6 = 14.25 \), which is a large value. For \( p(x)=4(1.5)^4+3=20.25 + 3 = 23.25 \). Wait, maybe better to use the y - intercept and HA.
- Top - right graph: Less steep ( \( a = 1 \) ), approaches \( y = 3 \) as \( x ightarrow-\infty \) (HA \( y = 3 \)), and at \( x = 0 \), \( t(0)=4 \). So this matches \( t(x)=1.5^x + 3 \) (HA \( y = 3 \), \( a = 1 \), \( t(0)=4 \)).
- Bottom - right graph: Less steep ( \( a = 1 \) ), approaches \( y=-6 \) as \( x ightarrow-\infty \) (HA \( y=-6 \)), and at \( x = 0 \), \( w(0)=-5 \). So this matches \( w(x)=1.5^x-6 \) (HA \( y=-6 \), \( a = 1 \), \( w(0)=-5 \)).
- Top - left graph: Steep ( \( a = 4 \) ), approaches \( y=-6 \) as \( x ightarrow-\infty \) (HA \( y=-6 \)), and at \( x = 0 \), \( m(0)=-2 \). So \( m(x)=4(1.5)^x-6 \).
- Bottom - left graph: Steep ( \( a = 4 \) ), approaches \( y = 3 \) as \( x ightarrow-\infty \) (HA \( y = 3 \)), and at \( x = 0 \), \( p(0)=7 \). So \( p(x)=4(1.5)^x + 3 \).

So:
- Top - left: \( m(x)=4(1.5)^x-6 \)
- Top - right: \( t(x)=1.5^x + 3 \)
- Bottom - left: \( p(x)=4(1.5)^x + 3 \)
- Bottom - right: \( w(x)=1.5^x-6 \)

Answer:

Top - left: \( \boldsymbol{m(x)=4(1.5)^x - 6} \)
Top - right: \( \boldsymbol{t(x)=1.5^x + 3} \)
Bottom - left: \( \boldsymbol{p(x)=4(1.5)^x + 3} \)
Bottom - right: \( \boldsymbol{w(x)=1.5^x - 6} \)