The problem requires us to use the given graph of the function $f(x) = x^2 - 3x - 4$ in the domain $-2 \le x \le 5$ to determine: (i) the values of $x$ for which $f(x) = 0$ (ii) the values of $x$ for which $f(x) = 6$ (iii) the values of $x$ for which $f(x) = -4$ (iv) the value of $f(2)$ (v) the coordinates of the minimum point of the curve (vi) the minimum value of $f(x)$

1. Problem Description

The problem requires us to use the given graph of the function $f(x) = x^2 - 3x - 4$ in the domain $-2 \le x \le 5$ to determine:

(i) the values of $x$ for which $f(x) = 0$

(ii) the values of $x$ for which $f(x) = 6$

(iii) the values of $x$ for which $f(x) = -4$

(iv) the value of $f(2)$

(v) the coordinates of the minimum point of the curve

(vi) the minimum value of $f(x)$

2. Solution Steps

(i) To find the values of $x$ for which $f(x) = 0$, we look for the $x$-intercepts of the graph. From the graph, the $x$-intercepts are $x = -1$ and $x = 4$.

(ii) To find the values of $x$ for which $f(x) = 6$, we look for the points on the graph where $y = 6$. From the graph, the corresponding $x$ values are approximately $x = -2$ and $x = 5$.

(iii) To find the values of $x$ for which $f(x) = -4$, we look for the points on the graph where $y = -4$. From the graph, $x=0$ and $x=3$.

(iv) To find the value of $f(2)$, we look for the point on the graph where $x = 2$. From the graph, the corresponding $y$ value is $f(2) = -6$.

(v) To find the coordinates of the minimum point of the curve, we look for the lowest point on the graph. From the graph, the minimum point appears to be at approximately $(1.5, -6.25)$.

(vi) To find the minimum value of $f(x)$, we look for the $y$-coordinate of the minimum point of the curve. From the graph, the minimum value of $f(x)$ is approximately $-6.25$.

3. Final Answer

(i) $x = -1, 4$

(ii) $x = -2, 5$

(iii) $x = 0, 3$

(iv) $f(2) = -6$

(v) $(1.5, -6.25)$

(vi) $-6.25$