1. Find the axis of symmetry, vertex, and $$ y $$-intercept of $$ f(x)=2 x^{2}-4 x-7 $$.Then graph the function.

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**Step 1. Find the Axis of Symmetry**

The axis of symmetry for a quadratic function in the form  
$$
f(x)=ax^2+bx+c
$$  
is given by  
$$
x=\frac{-b}{2a}.
$$

For the function  
$$
f(x)=2x^2-4x-7,
$$  
we have $$ a=2 $$ and $$ b=-4 $$. Thus,  
$$
x=\frac{-(-4)}{2(2)}=\frac{4}{4}=1.
$$

So, the axis of symmetry is $$ x=1 $$.

---

**Step 2. Find the Vertex**

The vertex lies on the axis of symmetry. To find the $$ y $$-coordinate of the vertex, substitute $$ x=1 $$ into $$ f(x) $$:

$$
f(1)=2(1)^2-4(1)-7=2-4-7=-9.
$$

Thus, the vertex is  
$$
(1,-9).
$$

---

**Step 3. Find the $$ y $$-Intercept**

The $$ y $$-intercept occurs when $$ x=0 $$. Evaluate $$ f(0) $$:

$$
f(0)=2(0)^2-4(0)-7=-7.
$$

Thus, the $$ y $$-intercept is  
$$
(0,-7).
$$

---

**Step 4. Graph the Function**

1. **Plot the Vertex and $$ y $$-Intercept:**  
   - Plot the vertex $$ (1, -9) $$.  
   - Plot the $$ y $$-intercept $$ (0, -7) $$.

2. **Axis of Symmetry:**  
   - Draw a vertical line at $$ x=1 $$; this is the axis of symmetry.

3. **Additional Points:**  
   - Choose a value for $$ x $$ on either side of the axis. For example, take $$ x=2 $$:
     $$
     f(2)=2(2)^2-4(2)-7=2(4)-8-7=8-8-7=-7.
     $$
     So, $$ (2,-7) $$ is a point on the graph.
   - Note: $$ (2,-7) $$ is symmetric to $$ (0,-7) $$ about the axis $$ x=1 $$.

4. **Sketch the Parabola:**  
   - Plot the points $$ (0,-7) $$, $$ (1,-9) $$, and $$ (2,-7) $$.  
   - Since $$ a=2>0 $$, the parabola opens upward.
   - Draw a smooth curve through these points making sure it is symmetric with respect to $$ x = 1 $$.

---

**Final Answer**

- **Axis of Symmetry:** $$ x=1 $$
- **Vertex:** $$ (1,-9) $$
- **$$ y $$-Intercept:** $$ (0,-7) $$

The graph is a parabola opening upward, symmetric about the line $$ x=1 $$, with the vertex at $$ (1,-9) $$ and passing through the points $$ (0,-7) $$ and $$ (2,-7) $$.
- **Axis of Symmetry:** $$ x=1 $$ - **Vertex:** $$ (1, -9) $$ - **$$ y $$-Intercept:** $$ (0, -7) $$ The graph is a parabola opening upward, symmetric about $$ x=1 $$, with the vertex at $$ (1, -9) $$ and passing through $$ (0, -7) $$ and $$ (2, -7) $$.

1. Find the axis of symmetry, vertex, and \( y \)-intercept of \( f(x)=2 x^{2}-4 x-7 \).Then graph the function.

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