Questions: For the following quadratic function, tell whether the graph opens upward or downward and whether the graph is wider, narrower, or the same as (f(x)=x^2). (F(x)=4 x^2+7) The graph opens A. upward. B. downward.

For the following quadratic function, tell whether the graph opens upward or downward and whether the graph is wider, narrower, or the same as (f(x)=x^2).
(F(x)=4 x^2+7)

The graph opens A. upward. B. downward.

Transcript text: For the following quadratic function, tell whether the graph opens upward or downward and whether the graph is wider, narrower, or the same as $f(x)=x^{2}$. \[ F(x)=4 x^{2}+7 \] The graph opens A. upward. B. downward.

Solution

Solution Steps

Step 1: Direction of Opening

The graph of the quadratic function $F(x) = 4x^2 + 0x + 7$ opens upward. This is determined by the sign of the coefficient $a$. If $a > 0$, the graph opens upward; if $a < 0$, it opens downward.

Step 2: Relative Width

The graph of $F(x) = 4x^2 + 0x + 7$ is narrower than the graph of $f(x) = x^2$. This is because the absolute value of $a$ affects the rate at which the function's value changes as $x$ moves away from the vertex. A larger $|a|$ means the graph is steeper (narrower), and a smaller $|a|$ means the graph is less steep (wider).