Question

the coordinate plane shows the graph of the quadratic function f(x). which of the following describes the graph of -3·f(x)?

Explanation:

Step1: Analyze transformation rules

For a function \( y = a \cdot f(x) \), if \( |a|>1 \), it's a vertical stretch; if \( 0<|a|<1 \), vertical compression. If \( a<0 \), reflect over x - axis. Here \( a=-3 \), so first reflect \( f(x) \) over x - axis, then vertically stretch by factor 3.

Step2: Analyze original graph

Original \( f(x) \) is a parabola opening up (since coefficient of \( x^{2} \) is positive), vertex at \( (0, - 9) \) (from graph, lowest point at y=-9), x - intercepts at \( x=-1 \) and \( x = 4 \)? Wait, no, looking at grid, x - intercepts at \( x=-1 \)? Wait, the original graph crosses x - axis at \( x=-1 \)? Wait, no, the grid: x from -16 to 16, y from -16 to 16. The original parabola: when x = -1, y=0? Wait, no, looking at the graph, the roots (x - intercepts) are at \( x=-1 \) and \( x = 4 \)? Wait, no, the graph crosses x - axis at x=-1? Wait, no, the left x - intercept: when x=-1, y=0? Wait, the grid lines: each square is 1 unit. So original \( f(x) \): vertex at (0, - 9), opens up, x - intercepts at x=-1 and x = 4? Wait, no, when x = -1, y=0; x = 4, y=0? Wait, no, let's check: when x=0, y=-9. So original \( f(x)=a(x - 0)^{2}-9=ax^{2}-9 \). When y = 0, \( ax^{2}-9 = 0\), \( x^{2}=\frac{9}{a} \). From graph, when x = 3, y=0? Wait, maybe I misread. Anyway, the key is transformation: \( - 3f(x) \) will reflect over x - axis (so opens down) and stretch vertically by 3. So vertex of original at (0, - 9), after reflection: (0, 9), then stretch by 3: (0, 27). X - intercepts remain same (since multiplying by - 3 doesn't change where y = 0, because \( - 3f(x)=0\) when \( f(x)=0 \)). The direction: original opens up, new opens down. Vertical stretch: each y - value is multiplied by - 3, so the graph is reflected over x - axis and vertically stretched by a factor of 3.

Answer:

The graph of \( - 3\cdot f(x) \) is a parabola that opens downward, is vertically stretched by a factor of 3 compared to \( f(x) \), has the same x - intercepts as \( f(x) \), and its vertex is at \( (0, 27) \) (since original vertex y - coordinate is - 9, \( - 3\times(-9)=27 \)) while being reflected over the x - axis (so opening direction changes from up to down).