Given $$ f(x)=-3 x^{2}+3 $$, sketch the graph of $$ f $$, clearly showing ALL intercepts with the axes and turning points. 1. Use your sketch to answer the following. INSTRUCTION: Give your answers as coordinates in the form $$ (\mathbf{x} ; \mathbf{y}) $$. Answer: 1. The $$ x $$-intercepts of $$ f $$ are and 2. The $$ y $$-intercept of $$ f $$ is 3. The graph of $$ f $$ is Select

Question

UpStudy AI · Accepted Answer

To analyze the function $$ f(x) = -3x^2 + 3 $$, we will follow these steps:

1. **Find the intercepts with the axes.**
2. **Determine the turning points.**
3. **Sketch the graph based on the findings.**

### Step 1: Find the intercepts

**1.1. Find the $$ x $$-intercepts:**
The $$ x $$-intercepts occur when $$ f(x) = 0 $$.

$$
-3x^2 + 3 = 0
$$

Rearranging gives:

$$
3x^2 = 3
$$

Dividing both sides by 3:

$$
x^2 = 1
$$

Taking the square root of both sides:

$$
x = \pm 1
$$

Thus, the $$ x $$-intercepts are $$ (1; 0) $$ and $$ (-1; 0) $$.

**1.2. Find the $$ y $$-intercept:**
The $$ y $$-intercept occurs when $$ x = 0 $$.

$$
f(0) = -3(0)^2 + 3 = 3
$$

Thus, the $$ y $$-intercept is $$ (0; 3) $$.

### Step 2: Determine the turning points

The function $$ f(x) = -3x^2 + 3 $$ is a quadratic function that opens downwards (since the coefficient of $$ x^2 $$ is negative). The vertex of the parabola, which is the turning point, can be found using the formula:

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

For our function, $$ a = -3 $$ and $$ b = 0 $$:

$$
x = -\frac{0}{2 \cdot -3} = 0
$$

Now, substituting $$ x = 0 $$ back into the function to find the $$ y $$-coordinate of the vertex:

$$
f(0) = 3
$$

Thus, the turning point (vertex) is $$ (0; 3) $$.

### Step 3: Sketch the graph

Now we can summarize the findings:

- $$ x $$-intercepts: $$ (1; 0) $$ and $$ (-1; 0) $$
- $$ y $$-intercept: $$ (0; 3) $$
- Turning point: $$ (0; 3) $$

### Final Answers

1. The $$ x $$-intercepts of $$ f $$ are $$ (1; 0) $$ and $$ (-1; 0) $$.
2. The $$ y $$-intercept of $$ f $$ is $$ (0; 3) $$.
3. The graph of $$ f $$ is a downward-opening parabola with its vertex at the $$ y $$-intercept.

The graph will look like a parabola opening downwards, intersecting the x-axis at $$ (1; 0) $$ and $$ (-1; 0) $$, and the y-axis at $$ (0; 3) $$.
The $$ x $$-intercepts are $$ (1, 0) $$ and $$ (-1, 0) $$, the $$ y $$-intercept is $$ (0, 3) $$, and the graph is a downward-opening parabola with its vertex at $$ (0, 3) $$.

Given , sketch the graph of ,
clearly showing ALL intercepts with the axes and
turning points.
1.
Use your sketch to answer the following.
INSTRUCTION: Give your answers as
coordinates in the form .
Answer:

The -intercepts of are
and

The -intercept of is

The graph of is
Select

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Pre Calculus Questions

Given , sketch the graph of , clearly showing ALL intercepts with the axes and turning points. 1. Use your sketch to answer the following. INSTRUCTION: Give your answers as coordinates in the form . Answer: The -intercepts of are and The -intercept of is The graph of is Select