Exponential Functions and Graphs Practice

Use the base and the direction together; do not classify a curve from its initial height.

The rendered graph comes from \(y=4(2)^{x}\). Its base is \(2\), so it shows exponential growth; the exact plotted points follow that same multiplier.

Use the base and the direction together; do not classify a curve from its initial height.

The rendered graph comes from \(y=18(0.5)^{x}\). Its base is \(0.5\), so it shows exponential decay; the exact plotted points follow that same multiplier.

Use the base and the direction together; do not classify a curve from its initial height.

The rendered graph comes from \(y=5(1.4)^{x}\). Its base is \(1.4\), so it shows exponential growth; the exact plotted points follow that same multiplier.

Use the base and the direction together; do not classify a curve from its initial height.

The rendered graph comes from \(y=24(0.75)^{x}\). Its base is \(0.75\), so it shows exponential decay; the exact plotted points follow that same multiplier.

Use the base and the direction together; do not classify a curve from its initial height.

The rendered graph comes from \(y=2(3)^{x}\). Its base is \(3\), so it shows exponential growth; the exact plotted points follow that same multiplier.

Set the exponent equal to zero; every positive base to the zero power equals one.

At \(x=0\), \(y=7(1.5)^0=7\). The graph labels this exact point, so the intercept is \((0,7)\).

Set the exponent equal to zero; every positive base to the zero power equals one.

At \(x=0\), \(y=30(0.8)^0=30\). The graph labels this exact point, so the intercept is \((0,30)\).

Set the exponent equal to zero; every positive base to the zero power equals one.

At \(x=0\), \(y=2(2.5)^0=2\). The graph labels this exact point, so the intercept is \((0,2)\).

Set the exponent equal to zero; every positive base to the zero power equals one.

At \(x=0\), \(y=16(0.6)^0=16\). The graph labels this exact point, so the intercept is \((0,16)\).

Set the exponent equal to zero; every positive base to the zero power equals one.

At \(x=0\), \(y=9(1.2)^0=9\). The graph labels this exact point, so the intercept is \((0,9)\).

Calculate the power before multiplying by the initial scale.

Substitute the displayed horizontal coordinate: \(f(3)=3(2)^{3}=24\). The plotted point is generated from that same value.

Calculate the power before multiplying by the initial scale.

Substitute the displayed horizontal coordinate: \(f(2)=40(0.5)^{2}=10\). The plotted point is generated from that same value.

Calculate the power before multiplying by the initial scale.

Substitute the displayed horizontal coordinate: \(f(2)=5(1.6)^{2}=12.8\). The plotted point is generated from that same value.

Calculate the power before multiplying by the initial scale.

Substitute the displayed horizontal coordinate: \(f(3)=81(0.3333333333333333)^{3}=3\). The plotted point is generated from that same value.

Calculate the power before multiplying by the initial scale.

Substitute the displayed horizontal coordinate: \(f(4)=8(1.25)^{4}=19.5313\). The plotted point is generated from that same value.

Locate the variable: exponential functions place it in the exponent of a constant base.

In \(6(1.3)^x\), the variable appears in the exponent and the base is positive and not \(1\). The other expressions place the variable outside the exponent or use a power of \(x\).

Locate the variable: exponential functions place it in the exponent of a constant base.

In \(14(0.8)^x\), the variable appears in the exponent and the base is positive and not \(1\). The other expressions place the variable outside the exponent or use a power of \(x\).

Locate the variable: exponential functions place it in the exponent of a constant base.

In \(2^x\), the variable appears in the exponent and the base is positive and not \(1\). The other expressions place the variable outside the exponent or use a power of \(x\).

Locate the variable: exponential functions place it in the exponent of a constant base.

In \(9(3)^x\), the variable appears in the exponent and the base is positive and not \(1\). The other expressions place the variable outside the exponent or use a power of \(x\).

Locate the variable: exponential functions place it in the exponent of a constant base.

In \(25(0.4)^x\), the variable appears in the exponent and the base is positive and not \(1\). The other expressions place the variable outside the exponent or use a power of \(x\).

Use x = 0 to identify the scale, then compare consecutive outputs to identify the factor.

Because \(q(0)=12(1.08)^0=12\), the initial output is \(12\). The base \(1.08\) is the repeated one-step multiplier.

Use x = 0 to identify the scale, then compare consecutive outputs to identify the factor.

Because \(q(0)=75(0.92)^0=75\), the initial output is \(75\). The base \(0.92\) is the repeated one-step multiplier.

Use x = 0 to identify the scale, then compare consecutive outputs to identify the factor.

Because \(q(0)=4(2.5)^0=4\), the initial output is \(4\). The base \(2.5\) is the repeated one-step multiplier.

Use x = 0 to identify the scale, then compare consecutive outputs to identify the factor.

Because \(q(0)=160(0.6)^0=160\), the initial output is \(160\). The base \(0.6\) is the repeated one-step multiplier.

Use x = 0 to identify the scale, then compare consecutive outputs to identify the factor.

Because \(q(0)=9(1.4)^0=9\), the initial output is \(9\). The base \(1.4\) is the repeated one-step multiplier.

Read a from the x = 0 row and verify a constant ratio across at least two row pairs.

The table gives \(f(0)=5\), so \(a=5\). Each output divided by the preceding output is \(2\), so \(b=2\) and the model is \(f(x)=5(2)^{x}\).

Read a from the x = 0 row and verify a constant ratio across at least two row pairs.

The table gives \(f(0)=64\), so \(a=64\). Each output divided by the preceding output is \(0.5\), so \(b=0.5\) and the model is \(f(x)=64(0.5)^{x}\).

Read a from the x = 0 row and verify a constant ratio across at least two row pairs.

The table gives \(f(0)=8\), so \(a=8\). Each output divided by the preceding output is \(1.5\), so \(b=1.5\) and the model is \(f(x)=8(1.5)^{x}\).

Read a from the x = 0 row and verify a constant ratio across at least two row pairs.

The table gives \(f(0)=125\), so \(a=125\). Each output divided by the preceding output is \(0.2\), so \(b=0.2\) and the model is \(f(x)=125(0.2)^{x}\).

Read a from the x = 0 row and verify a constant ratio across at least two row pairs.

The table gives \(f(0)=20\), so \(a=20\). Each output divided by the preceding output is \(1.25\), so \(b=1.25\) and the model is \(f(x)=20(1.25)^{x}\).

Evaluate both models at the same input before comparing; visual height alone is not sufficiently precise.

From the exact functions, Model A gives \(54.88\) and Model B gives \(48.384\). Their absolute difference is \(6.496\), consistent with the plotted points.

Evaluate both models at the same input before comparing; visual height alone is not sufficiently precise.

From the exact functions, Model A gives \(28.8\) and Model B gives \(24.3\). Their absolute difference is \(4.5\), consistent with the plotted points.

Evaluate both models at the same input before comparing; visual height alone is not sufficiently precise.

From the exact functions, Model A gives \(78.643\) and Model B gives \(73.242\). Their absolute difference is \(5.401\), consistent with the plotted points.

Evaluate both models at the same input before comparing; visual height alone is not sufficiently precise.

From the exact functions, Model A gives \(27.44\) and Model B gives \(36.847\). Their absolute difference is \(9.408\), consistent with the plotted points.

Evaluate both models at the same input before comparing; visual height alone is not sufficiently precise.

From the exact functions, Model A gives \(48\) and Model B gives \(50.625\). Their absolute difference is \(2.625\), consistent with the plotted points.

Translate the requested comparison into exact endpoint values before using the graph as a consistency check.

Over \([0,2]\), the average rates are \(15\) for Model A and \(11.25\) for Model B. Comparing values identifies Model A.

Translate the requested comparison into exact endpoint values before using the graph as a consistency check.

At \(x=3\), Model A is \(25.6\) and Model B is \(13.72\), so Model B has the smaller final value.

Translate the requested comparison into exact endpoint values before using the graph as a consistency check.

At \(x=0\), Model A is \(16\) and Model B is \(20\). Relative to Model A, the difference is \(\frac{20-16}{16}\cdot100=25\%\).

Translate the requested comparison into exact endpoint values before using the graph as a consistency check.

Over \([0,3]\), the average rates are \(-23.52\) for Model A and \(-11.563\) for Model B. Comparing magnitudes identifies Model A.

Translate the requested comparison into exact endpoint values before using the graph as a consistency check.

At \(x=3\), Model A is \(46.656\) and Model B is \(43.94\), so Model A has the larger final value.

Verify intercept, base classification, and one ratio as separate graph facts.

Its domain is all real numbers, its positive-scale range is positive, and it increases. This follows from \(h(0)=6\), the base \(1.7\), and the exact values used to render the curve.

Verify intercept, base classification, and one ratio as separate graph facts.

It passes through (0, 50), stays positive, and decreases toward zero as x increases. This follows from \(h(0)=50\), the base \(0.65\), and the exact values used to render the curve.

Verify intercept, base classification, and one ratio as separate graph facts.

Its one-step output ratio is 2.2, not a constant additive difference. This follows from \(h(0)=3\), the base \(2.2\), and the exact values used to render the curve.

Verify intercept, base classification, and one ratio as separate graph facts.

It retains 90% per step, so the graph shows gradual exponential decay. This follows from \(h(0)=100\), the base \(0.9\), and the exact values used to render the curve.

Verify intercept, base classification, and one ratio as separate graph facts.

Its vertical-axis intercept is (0, 1), and multiplying x by nothing is not the growth rule. This follows from \(h(0)=1\), the base \(3\), and the exact values used to render the curve.

Check x = 0, the base interval, and the axis scale before accepting a graph claim.

The 12 is the initial output; 1.4 is the one-step growth factor. The equation, exact values, and graph must describe the same initial output and repeated multiplier.

Check x = 0, the base interval, and the axis scale before accepting a graph claim.

The base 0.82 lies between 0 and 1, so the function shows decay. The equation, exact values, and graph must describe the same initial output and repeated multiplier.

Check x = 0, the base interval, and the axis scale before accepting a graph claim.

Repeated percentage decrease has a constant ratio, so its graph is exponential, not linear. The equation, exact values, and graph must describe the same initial output and repeated multiplier.

Check x = 0, the base interval, and the axis scale before accepting a graph claim.

Since 2^0=1, f(0)=7 and the intercept is (0, 7). The equation, exact values, and graph must describe the same initial output and repeated multiplier.

Check x = 0, the base interval, and the axis scale before accepting a graph claim.

Graph steepness must be compared on the stated scales or through exact values at common inputs. The equation, exact values, and graph must describe the same initial output and repeated multiplier.