Consider the parabola given by the equation:

Find the following for this parabola:
A) Does the graph of the resulting parabola open up or down?
B) The vertex is at the point

Question

Consider the parabola given by the equation: $$ f(x)=-3 x^{2}-6 x+4 $$ Find the following for this parabola: A) Does the graph of the resulting parabola open up or down? B) The vertex is at the point

UpStudy AI · Accepted Answer

To analyze the given parabola $$ f(x) = -3x^2 - 6x + 4 $$, we can answer the questions as follows:

### A) Does the graph of the resulting parabola open up or down?

The coefficient of the $$ x^2 $$ term in the quadratic equation $$ f(x) = ax^2 + bx + c $$ determines the direction in which the parabola opens. Here, $$ a = -3 $$.

- If $$ a > 0 $$, the parabola opens upwards.
- If $$ a < 0 $$, the parabola opens downwards.

Since $$ a = -3 < 0 $$, the graph of the parabola opens **down**.

### B) The vertex is at the point

The vertex of a parabola given by the equation $$ f(x) = ax^2 + bx + c $$ can be found using the formula for the x-coordinate of the vertex:

$$
x = -\frac{b}{2a}
$$

In our case, $$ a = -3 $$ and $$ b = -6 $$. Plugging in these values:

$$
x = -\frac{-6}{2 \cdot -3} = \frac{6}{-6} = -1
$$

Now, we can find the y-coordinate of the vertex by substituting $$ x = -1 $$ back into the function $$ f(x) $$:

$$
f(-1) = -3(-1)^2 - 6(-1) + 4
$$
$$
= -3(1) + 6 + 4
$$
$$
= -3 + 6 + 4 = 7
$$

Thus, the vertex of the parabola is at the point $$ (-1, 7) $$.

### Summary:
A) The graph of the parabola opens **down**.  
B) The vertex is at the point $$ (-1, 7) $$.
A) The graph opens down. B) The vertex is at (-1, 7).

Consider the parabola given by the equation:

Find the following for this parabola:
A) Does the graph of the resulting parabola open up or down?
B) The vertex is at the point

Solución de inteligencia artificial de Upstudy

Responder

Solución

A) Does the graph of the resulting parabola open up or down?

B) The vertex is at the point

Summary:

Beyond the Answer

preguntas relacionadas

Latest Pre Calculus Questions

Consider the parabola given by the equation: Find the following for this parabola: A) Does the graph of the resulting parabola open up or down? B) The vertex is at the point

Solución de inteligencia artificial de Upstudy

Responder

Solución

A) Does the graph of the resulting parabola open up or down?

B) The vertex is at the point

Summary:

Beyond the Answer

preguntas relacionadas

Latest Pre Calculus Questions

Consider the parabola given by the equation:

Find the following for this parabola:
A) Does the graph of the resulting parabola open up or down?
B) The vertex is at the point