Find the equations of the following functions:

a) To find the equation of the straight line, we first calculate the slope ($m$) using the two given points $(-2, -1)$ and $(1, 4)$.

Step 1: Calculate the slope ($m$). The formula for the slope is $m = \frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1, y_1) = (-2, -1)$ and $(x_2, y_2) = (1, 4)$. $m = \frac{4 - (-1)}{1 - (-2)} = \frac{4 + 1}{1 + 2} = \frac{5}{3}$

Step 2: Use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, or the slope-intercept form, $y = mx + c$. We will use the slope-intercept form. Substitute the slope $m = \frac{5}{3}$ and one of the points, for example $(1, 4)$, into $y = mx + c$. $4 = \frac{5}{3}(1) + c$ $4 = \frac{5}{3} + c$ $c = 4 - \frac{5}{3}$ $c = \frac{12}{3} - \frac{5}{3}$ $c = \frac{7}{3}$

Step 3: Write the equation of the line. Substitute $m = \frac{5}{3}$ and $c = \frac{7}{3}$ into $y = mx + c$. $y = \frac{5}{3}x + \frac{7}{3}$ The equation of the line is $\boxed{y = \frac{5}{3}x + \frac{7}{3}}$.

b) To find the equation of the parabola, we use the vertex form of a quadratic function, $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

Step 1: Identify the vertex and substitute it into the vertex form. From the graph, the vertex is $(h, k) = (4, -1)$. $y = a(x - 4)^2 - 1$

Step 2: Use another point on the parabola to find the value of $a$. From the graph, the parabola appears to pass through the point $(5, 0)$ on the x-axis. Substitute $(x, y) = (5, 0)$ into the equation. $0 = a(5 - 4)^2 - 1$ $0 = a(1)^2 - 1$ $0 = a - 1$ $a = 1$

Step 3: Write the equation of the parabola. Substitute $a = 1$ into the vertex form. $y = 1(x - 4)^2 - 1$ $y = (x - 4)^2 - 1$ The equation of the parabola is $\boxed{y = (x - 4)^2 - 1}$.

Send me the next one 📸