Given: \( h(x)=2^{x}-4 \) \( 1.1 \quad \) Write down the equation of asymptote. 1.2 Determine the coordinates of the \( y \)-intercept of \( h \). 1.3 Calculate the \( x \)-intercept of \( h \). \( 1.4 \quad \) Draw a sketch graph of \( h \), clearly showing all asymptotes, intercepts with the axes and at least one other point on \( h \). \( 1.5 \quad \) Write down range of \( h \). 1.6 Describe the transformation from \( h \) to \( g \) if \( g(x)=2^{-x}+1 \)

Let's solve the problem step by step. ### 1.1 Write down the equation of asymptote. The function \( h(x) = 2^x - 4 \) is an exponential function. The horizontal asymptote occurs as \( x \) approaches negative infinity. As \( x \to -\infty \), \( 2^x \to 0 \), thus: \[ h(x) \to 0 - 4 = -4 \] Therefore, the equation of the asymptote is: \[ y = -4 \] ### 1.2 Determine the coordinates of the \( y \)-intercept of \( h \). To find the \( y \)-intercept, we evaluate \( h(0) \): \[ h(0) = 2^0 - 4 = 1 - 4 = -3 \] Thus, the coordinates of the \( y \)-intercept are: \[ (0, -3) \] ### 1.3 Calculate the \( x \)-intercept of \( h \). To find the \( x \)-intercept, we set \( h(x) = 0 \): \[ 2^x - 4 = 0 \] Solving for \( x \): \[ 2^x = 4 \] Since \( 4 = 2^2 \), we have: \[ x = 2 \] Thus, the coordinates of the \( x \)-intercept are: \[ (2, 0) \] ### 1.4 Draw a sketch graph of \( h \). To sketch the graph of \( h(x) = 2^x - 4 \), we note the following points: - Asymptote: \( y = -4 \) - \( y \)-intercept: \( (0, -3) \) - \( x \)-intercept: \( (2, 0) \) We can also calculate another point, for example, \( h(1) \): \[ h(1) = 2^1 - 4 = 2 - 4 = -2 \] So, another point is \( (1, -2) \). Now, I will create a sketch graph of \( h \) with the identified points and asymptote. ### 1.5 Write down the range of \( h \). The function \( h(x) = 2^x - 4 \) approaches the asymptote \( y = -4 \) as \( x \to -\infty \) and increases without bound as \( x \to \infty \). Therefore, the range of \( h \) is: \[ (-4, \infty) \] ### 1.6 Describe the transformation from \( h \) to \( g \) if \( g(x) = 2^{-x} + 1 \). The function \( g(x) = 2^{-x} + 1 \) can be analyzed as follows: - The term \( 2^{-x} \) is a reflection of \( 2^x \) across the \( y \)-axis. - The addition of \( 1 \) translates the entire graph of \( 2^{-x} \) upward by \( 1 \) unit. Thus, the transformation from \( h \) to \( g \) involves: 1. Reflection across the \( y \)-axis. 2. Vertical translation upward by \( 1 \) unit. Now, I will create a sketch graph of \( h \) based on the points and asymptote identified. ### Sketch Graph The graph will show: - The horizontal asymptote at \( y = -4 \). - The \( y \)-intercept at \( (0, -3) \). - The \( x \)-intercept at \( (2, 0) \). - The additional point \( (1, -2) \). Let's proceed to create the sketch graph. (Note: As a text-based AI, I cannot create visual sketches directly. However, you can plot the points and asymptote on graph paper or using graphing software to visualize the function.)