1. **Problem Statement:** We are given two functions to analyze and graph: - Function 1: $y = x^3$ - Function 2: $y = -x + 1$ 2. **Function 1: $y = x^3$** - This is a cubic function. - The general form of a cubic function is $y = ax^3 + bx^2 + cx + d$; here $a=1$, $b=0$, $c=0$, $d=0$. - Important properties: - It is an odd function, symmetric about the origin. - The graph passes through the origin $(0,0)$. - As $x$ increases, $y$ increases rapidly; as $x$ decreases, $y$ decreases rapidly. 3. **Intermediate Work for $y = x^3$:** - Calculate some points: - $x=-2 \Rightarrow y=(-2)^3 = -8$ - $x=-1 \Rightarrow y=(-1)^3 = -1$ - $x=0 \Rightarrow y=0^3 = 0$ - $x=1 \Rightarrow y=1^3 = 1$ - $x=2 \Rightarrow y=2^3 = 8$ 4. **Function 2: $y = -x + 1$** - This is a linear function with slope $m = -1$ and y-intercept $b = 1$. - The general form is $y = mx + b$. - Important properties: - The line crosses the y-axis at $(0,1)$. - The slope $-1$ means the line decreases by 1 unit in $y$ for every 1 unit increase in $x$. 5. **Intermediate Work for $y = -x + 1$:** - Calculate some points: - $x=0 \Rightarrow y = -0 + 1 = 1$ - $x=1 \Rightarrow y = -1 + 1 = 0$ - $x=2 \Rightarrow y = -2 + 1 = -1$ 6. **Summary:** - The cubic function $y = x^3$ has a characteristic S-shaped curve passing through points $(-2,-8), (-1,-1), (0,0), (1,1), (2,8)$. - The linear function $y = -x + 1$ is a straight line crossing the y-axis at 1 and decreasing with slope $-1$. **Final answers:** - For the cubic function: $y = x^3$ - For the linear function: $y = -x + 1$