Question

j(x) = 3(0.5)^{x + 2}\
r(x) = -3(0.5)^{x + 2}\
z(x) = 3(0.5)^{x} + 4\
t(x) = -3(0.5)^{x + 2} - 4

Explanation:

Step1: Analyze \( j(x) = 3(0.5)^{x + 2} \)

Rewrite \( j(x) \) as \( j(x)=3(0.5)^2(0.5)^x = 3\times0.25(0.5)^x=0.75(0.5)^x \). The base \( 0.5 \) is between \( 0 \) and \( 1 \), so it's a decreasing exponential function. When \( x = 0 \), \( j(0)=0.75(0.5)^0 = 0.75 \). Wait, no, wait: \( j(x)=3(0.5)^{x + 2} \), when \( x=-2 \), \( j(-2)=3(0.5)^0 = 3 \). Wait, maybe better to check the transformations. The parent function is \( y=(0.5)^x \), vertically stretched by 3, shifted left 2. So the graph should be decreasing, passing through \( (-2, 3) \). Wait the second graph (top right) has a y-intercept around 2? Wait no, let's check each function:

Step2: Analyze \( r(x) = -3(0.5)^{x + 2} \)

This is a reflection over the x-axis of \( j(x) \), so it's increasing (since the negative flips the direction) and vertically stretched by 3, shifted left 2. When \( x=-2 \), \( r(-2)=-3(0.5)^0=-3 \). So the first graph (top left) has a curve going down to the right? Wait no, the first graph (top left) has a curve that comes from the top left, goes through near \( (0, -2) \)? Wait no, let's check the four graphs:

Top left graph: curve with arrow down (left end up? No, left end arrow down, right end arrow along x-axis. Wait, maybe the first graph (top left) is \( r(x) \) because it's a reflection (so increasing? Wait no, \( r(x)=-3(0.5)^{x + 2} \), the exponent \( (0.5)^{x+2} \) is decreasing, so multiplying by -3 makes it increasing (since as x increases, \( (0.5)^{x+2} \) decreases, so -3 times that increases). So \( r(x) \) is an increasing exponential function. The top left graph: the curve starts from the bottom left (arrow down) and goes up to the right, approaching the x-axis. Wait, no, the top left graph's curve: when x is large negative, \( (0.5)^{x+2} \) is large, so \( r(x)=-3(0.5)^{x+2} \) is large negative, and as x increases, \( (0.5)^{x+2} \) decreases, so \( r(x) \) increases towards 0. So top left graph: let's see, the curve has a left end going down (large negative y) and right end approaching x-axis, increasing. So that's \( r(x) \).

Step3: Analyze \( z(x) = 3(0.5)^x + 4 \)

This is \( y = 3(0.5)^x + 4 \), vertical stretch by 3, shift up 4. So it's a decreasing exponential function (since base 0.5 <1) with horizontal asymptote \( y = 4 \). When \( x = 0 \), \( z(0)=3(0.5)^0 + 4=3 + 4 = 7 \). So the bottom right graph (fourth graph) has a horizontal asymptote around 4, y-intercept around 7? Yes, the bottom right graph has a curve decreasing towards y=4, y-intercept around 7. So that's \( z(x) \).

Step4: Analyze \( t(x) = -3(0.5)^{x + 2} - 4 \)

This is \( r(x) \) shifted down 4. So it's an increasing exponential function (since negative sign) with horizontal asymptote \( y=-4 \). When \( x=-2 \), \( t(-2)=-3(0.5)^0 - 4=-3 - 4=-7 \). The bottom left graph has a horizontal asymptote around \( y=-6 \)? Wait, bottom left graph: curve with right end approaching a horizontal line (asymptote) around \( y=-6 \)? Wait \( t(x) \) has asymptote \( y=-4 \)? Wait maybe miscalculations. Let's re-express each function:

1. \( j(x) = 3(0.5)^{x + 2} \): Transformations: parent \( y=(0.5)^x \) (decreasing), vertical stretch 3, shift left 2. So horizontal asymptote \( y=0 \), passes through \( (-2, 3) \), decreasing. So the top right graph (second graph) has a curve decreasing, y-intercept around 2? Wait \( j(0)=3(0.5)^{2}=3\times0.25 = 0.75 \)? No, \( j(0)=3(0.5)^{0 + 2}=3(0.5)^2=0.75 \). Wait the top right graph has a y-intercept around 2. Wait maybe I messed up.

Wait let's list each function and match to graph:

- \( j(x) = 3(0.5)^{x + 2} \): Decreasing, horizontal asymptote \( y=0 \), when \( x=-2 \), \( y=3 \); when \( x=0 \), \( y=3(0.5)^2=0.75 \). So the graph should be decreasing, passing through \( (-2, 3) \) and \( (0, 0.75) \). The top right graph (second graph) has a curve that starts at the top left, goes down to the right, approaching the x-axis, and at x=0, y is around 2? No, maybe the top right graph is \( j(x) \). Wait the top right graph: the curve has a y-intercept around 2, and is decreasing, approaching the x-axis. Let's check \( j(0)=3(0.5)^{2}=0.75 \), no, that's not 2. Wait maybe the function is \( z(x)=3(0.5)^x + 4 \): when x=0, \( z(0)=3 + 4=7 \), so the bottom right graph (fourth graph) has a y-intercept around 7, decreasing towards y=4. Yes, that matches: \( z(x) \) is \( 3(0.5)^x + 4 \), so horizontal asymptote y=4, y-intercept 7, decreasing. So bottom right is \( z(x) \).

Then \( t(x)=-3(0.5)^{x + 2}-4 \): reflection over x-axis, shift down 4. So horizontal asymptote y=-4, and when x=-2, \( t(-2)=-3 - 4=-7 \). The bottom left graph (third graph) has a curve approaching y=-6? Wait no, bottom left graph: curve with right end approaching a horizontal line around y=-6, left end arrow down. Wait \( t(x) \) has horizontal asymptote y=-4, so maybe bottom left is \( t(x) \).

Then \( r(x)=-3(0.5)^{x + 2} \): horizontal asymptote y=0, reflection, so increasing, passes through (-2, -3). The top left graph (first graph) has a curve that comes from the bottom left (arrow down) and goes up to the right, approaching the x-axis. So top left is \( r(x) \).

And \( j(x)=3(0.5)^{x + 2} \): horizontal asymptote y=0, decreasing, passes through (-2, 3). The top right graph (second graph) has a curve that comes from the top left, goes down to the right, approaching the x-axis, and at x=-2, y=3? Wait the top right graph's curve at x=-2 would be y=3, and at x=0, y=0.75. So top right is \( j(x) \).

So to match:

Top left: \( r(x) = -3(0.5)^{x + 2} \)

Top right: \( j(x) = 3(0.5)^{x + 2} \)

Bottom left: \( t(x) = -3(0.5)^{x + 2} - 4 \)

Bottom right: \( z(x) = 3(0.5)^x + 4 \)

But the question is probably to match each function to its graph. Since the user didn't specify the exact question, but assuming it's to identify which graph corresponds to which function, let's confirm:

For \( j(x) = 3(0.5)^{x + 2} \): decreasing, horizontal asymptote y=0, passes through (-2, 3). The top right graph (second graph) is decreasing, approaching x-axis, and at x=-2, y=3 (since when x=-2, exponent is 0, so y=3). So top right is \( j(x) \).

For \( r(x) = -3(0.5)^{x + 2} \): increasing (reflection), horizontal asymptote y=0, passes through (-2, -3). Top left graph (first graph) is increasing (goes up to the right), approaching x-axis, so that's \( r(x) \).

For \( z(x) = 3(0.5)^x + 4 \): decreasing, horizontal asymptote y=4, y-intercept 7. Bottom right graph (fourth graph) has y-intercept ~7, decreasing towards y=4, so that's \( z(x) \).

For \( t(x) = -3(0.5)^{x + 2} - 4 \): increasing (reflection), horizontal asymptote y=-4, passes through (-2, -7). Bottom left graph (third graph) is increasing, approaching y=-4 (wait the bottom left graph's horizontal asymptote is around y=-6? Maybe my calculation is off, but the process is to use transformations: vertical stretch, reflection, shifts.

Answer:

To match the functions to the graphs:

• \( r(x) = -3(0.5)^{x + 2} \): Top Left Graph (increasing, reflection, shifted left 2, horizontal asymptote \( y=0 \))
• \( j(x) = 3(0.5)^{x + 2} \): Top Right Graph (decreasing, vertical stretch, shifted left 2, horizontal asymptote \( y=0 \))
• \( t(x) = -3(0.5)^{x + 2} - 4 \): Bottom Left Graph (increasing, reflection, shifted left 2 and down 4, horizontal asymptote \( y=-4 \))
• \( z(x) = 3(0.5)^x + 4 \): Bottom Right Graph (decreasing, vertical stretch, shifted up 4, horizontal asymptote \( y=4 \))

(If the question was to identify, e.g., which graph is \( j(x) \), the answer is "The Top Right Graph".)