1. The problem is to graph the functions: 1. $f(x) = 2x + 1$ 2. $g(x) = x^2 + 3$ 3. $h(x) = x - 21$ 4. $c(x) = \sqrt{x}$ 2. Let's analyze each function and understand their shapes: - $f(x) = 2x + 1$ is a linear function with slope 2 and y-intercept 1. - $g(x) = x^2 + 3$ is a quadratic function (parabola) opening upwards, shifted up by 3 units. - $h(x) = x - 21$ is a linear function with slope 1 and y-intercept -21. - $c(x) = \sqrt{x}$ is a square root function defined for $x \geq 0$, starting at the origin. 3. Important rules: - Linear functions graph as straight lines. - Quadratic functions graph as parabolas. - Square root functions graph as curves starting at $x=0$ and increasing slowly. 4. Intermediate work for each function: - For $f(x)$: - When $x=0$, $f(0) = 2(0) + 1 = 1$ (y-intercept). - When $x=1$, $f(1) = 2(1) + 1 = 3$. - For $g(x)$: - When $x=0$, $g(0) = 0^2 + 3 = 3$ (vertex). - When $x=1$, $g(1) = 1 + 3 = 4$. - When $x=-1$, $g(-1) = 1 + 3 = 4$. - For $h(x)$: - When $x=0$, $h(0) = 0 - 21 = -21$ (y-intercept). - When $x=21$, $h(21) = 21 - 21 = 0$ (x-intercept). - For $c(x)$: - Defined only for $x \geq 0$. - When $x=0$, $c(0) = 0$. - When $x=1$, $c(1) = 1$. - When $x=4$, $c(4) = 2$. 5. To graph these functions, plot the points calculated and draw the corresponding shapes: - $f(x)$: straight line through points $(0,1)$ and $(1,3)$. - $g(x)$: parabola with vertex at $(0,3)$ passing through $(1,4)$ and $(-1,4)$. - $h(x)$: straight line through $(0,-21)$ and $(21,0)$. - $c(x)$: curve starting at $(0,0)$ passing through $(1,1)$ and $(4,2)$. Final answer: The graphs of the functions are as described above, showing linear, quadratic, and square root shapes accordingly.