Sketch the graph of the quadratic function. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.)
f(x) = x2 + x − 7
Indicate the coordinates of the vertex, the y-intercept, and the x-intercepts (if any). (If an answer does not exist, enter DNE. If there are multiple x-intercepts, enter your answers as a comma-separated list.)
vertex	(x, y)	
= 

  
Correct: Your answer is correct.
 

y-intercept	y	=	  
Correct: Your answer is correct.
x-intercepts	x	=	  
Incorrect: Your answer is incorrect.
Since the graph is symmetric around the vertical line through the vertex, reflecting the y-intercept across the line of symmetry, we find the additional point 
(x, y) = 

    

 on the graph.
Determine the left-hand and right-hand behavior of the graph. (If an answer does not exist, enter DNE. If you need to use  or –, enter INFINITY or –INFINITY, respectively.)
As the graph goes to the left without bound, the values of y approach    .
As the graph goes to the right without bound, the values of y approach    .

Let's break this down step by step for the quadratic function:

\[
f(x) = x^2 + x - 7
\]

### 1. **Vertex:**
The vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:

\[
x = \frac{-b}{2a}
\]

Here, \( a = 1 \) and \( b = 1 \). So,

\[
x = \frac{-1}{2(1)} = -\frac{1}{2}
\]

Now, substitute \( x = -\frac{1}{2} \) into the equation to find the y-coordinate:

\[
f\left(-\frac{1}{2}\right) = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) - 7
\]

\[
f\left(-\frac{1}{2}\right) = \frac{1}{4} - \frac{1}{2} - 7 = -\frac{27}{4}
\]

Thus, the vertex is at:

\[
\left(-\frac{1}{2}, -\frac{27}{4}\right)
\]

### 2. **y-intercept:**
The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the function:

\[
f(0) = 0^2 + 0 - 7 = -7
\]

So, the y-intercept is at:

\[
(0, -7)
\]

### 3. **x-intercepts:**
To find the x-intercepts, set \( f(x) = 0 \):

\[
x^2 + x - 7 = 0
\]

Use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute \( a = 1 \), \( b = 1 \), and \( c = -7 \):

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-7)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 28}}{2} = \frac{-1 \pm \sqrt{29}}{2}
\]

So, the x-intercepts are:

\[
x = \frac{-1 + \sqrt{29}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{29}}{2}
\]

These are approximate values:

\[
x \approx 2.19 \quad \text{and} \quad x \approx -3.19
\]

### 4. **Additional point using symmetry:**
The graph is symmetric around the vertical line \( x = -\frac{1}{2} \). Since we know the y-intercept is at \( (0, -7) \), reflect this point across \( x = -\frac{1}{2} \). The distance between \( x = 0 \) and \( x = -\frac{1}{2} \) is \( \frac{1}{2} \), so we move \( \frac{1}{2} \) units to the left:

\[
x = -1
\]

Now, substitute \( x = -1 \) into the function to find the corresponding y-value:

\[
f(-1) = (-1)^2 + (-1) - 7 = 1 - 1 - 7 = -7
\]

So, the additional point is:

\[
(-1, -7)
\]

### 5. **Left-hand and right-hand behavior:**
For large positive or negative \( x \), the quadratic term \( x^2 \) dominates the behavior of the function.

- As \( x \to -\infty \), \( f(x) \to +\infty \).
- As \( x \to +\infty \), \( f(x) \to +\infty \).

### Summary of Key Points:
- **Vertex**: \( \left(-\frac{1}{2}, -\frac{27}{4}\right) \)
- **y-intercept**: \( (0, -7) \)
- **x-intercepts**: \( \frac{-1 + \sqrt{29}}{2}, \frac{-1 - \sqrt{29}}{2} \approx 2.19, -3.19 \)
- **Additional Point**: \( (-1, -7) \)
- **Left-hand behavior**: As \( x \to -\infty \), \( f(x) \to +\infty \).
- **Right-hand behavior**: As \( x \to +\infty \), \( f(x) \to +\infty \).

---

Would you like me to go over any specific part in more detail?  
Here are some related questions:

1. How do you find the axis of symmetry for a quadratic function?
2. What determines the direction of a parabola's opening?
3. How do you use the discriminant to find the number of x-intercepts?
4. How do you complete the square to find the vertex form of a quadratic function?
5. Can the quadratic formula ever give complex solutions?

**Tip**: The coefficient of the quadratic term (\(a\)) determines whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

Sketch the graph of the quadratic function. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) f(x) = x^2 + x − 7. Indicate the coordinates of the vertex, the y-intercept, and the x-intercepts (if any). (If an answer does not exist, enter DNE. If there are multiple x-intercepts, enter your answers as a comma-separated list.)

Learn how to graph the quadratic function f(x) = x^2 + x − 7 by identifying its vertex, y-intercept, and x-intercepts. Understand key algebraic concepts, including using the vertex and quadratic formulas, and explore parabola symmetry. Suitable for students in Grades 9-11.

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