Consider the following function, $$ f(x)=-\left(x^{2}+2 x-3\right) $$ Use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $$ x $$-intercept(s), Then check your results algebraically by writing the quadratic function in standard form. (If an answer does not exist, enter DNE, vertex $$ (x, f(x))=(\square $$ axis of symmetry $$ x $$-intercept smaller $$ x $$-value $$ \quad(x, f(x))=(\square $$ larger $$ x $$-value standard form $$ f(x, f(x))=\square $$

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**Step 1. Write the function in expanded form.**

We start with  
$$
f(x) = -\left(x^2+2x-3\right)
$$  
Distribute the negative sign to obtain  
$$
f(x) = -x^2-2x+3.
$$

**Step 2. Express the function in standard form by completing the square.**

Begin by factoring out $$-1$$ from the quadratic terms:
$$
f(x) = -\left(x^2+2x\right)+3.
$$

Complete the square for the expression $$x^2+2x$$. Recall that
$$
x^2+2x = \left(x+1\right)^2-1.
$$
Substitute back:
$$
f(x) = -\left[\left(x+1\right)^2-1\right]+3.
$$
Distribute the negative:
$$
f(x) = -(x+1)^2+1+3,
$$
and combine constants:
$$
f(x)=-(x+1)^2+4.
$$

Thus, the standard form is  
$$
f(x)=-(x+1)^2+4.
$$

**Step 3. Identify the vertex.**

The standard form  
$$
f(x)=a(x-h)^2+k
$$
reveals the vertex at $$ (h,k) $$. Here,  
$$
h=-1 \quad \text{and} \quad k=4.
$$
Thus, the vertex is  
$$
(-1,4).
$$

**Step 4. Determine the axis of symmetry.**

The axis of symmetry is the vertical line through the vertex:
$$
x=-1.
$$

**Step 5. Find the $$ x $$-intercepts.**

Set $$ f(x)=0 $$:
$$
-(x+1)^2+4=0.
$$
Solve for $$ (x+1)^2 $$:
$$
-(x+1)^2 = -4 \quad \Longrightarrow \quad (x+1)^2=4.
$$
Taking the square root on both sides:
$$
x+1=\pm 2.
$$
Solve for $$ x $$:
- For the positive root: $$ x+1=2 $$ leads to $$ x=1 $$.
- For the negative root: $$ x+1=-2 $$ leads to $$ x=-3 $$.

Thus, the $$ x $$-intercepts are:
- Smaller $$ x $$-value: $$(-3,0)$$.
- Larger $$ x $$-value: $$(1,0)$$.

**Summary of Results**

- **Vertex:** $$(-1,4)$$
- **Axis of symmetry:** $$x=-1$$
- **$$ x $$-intercepts:**
  - Smaller $$ x $$-value: $$(-3,0)$$
  - Larger $$ x $$-value: $$(1,0)$$
- **Standard form:**  
  $$
  f(x)=-(x+1)^2+4.
  $$
**Summary of Results** - **Vertex:** $$(-1, 4)$$ - **Axis of Symmetry:** $$x = -1$$ - **$$ x $$-Intercepts:** - Smaller $$ x $$-value: $$(-3, 0)$$ - Larger $$ x $$-value: $$(1, 0)$$ - **Standard Form:** $$ f(x) = -(x + 1)^2 + 4 $$

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