Answer the questions about the following function.
f left parenthesis x right parenthesis equals 2 x squared minus x minus 1
​(a)  Is the point left parenthesis negative 2 comma 9 right parenthesis on the graph of​ f?
​(b)  If x​ = 2​, what is​ f(x)? What point is on the graph of​ f?
​(c)  If​ f(x) = negative 1​, what is​ x? What​ point(s) are on the graph of​ f?
​(d)  What is the domain of​ f?
​(e)  List the​ x-intercept(s), if​ any, of the graph of f.
​(f)  List the​ y-intercept, if​ any, of the graph of f.

We are given the quadratic function \( f(x) = 2x^2 - x - 1 \), and we'll answer each part of the question accordingly.

### (a) Is the point \((-2, 9)\) on the graph of \(f\)?
To check if the point \((-2, 9)\) is on the graph of the function, substitute \(x = -2\) into \(f(x)\) and see if it equals 9.

\[
f(-2) = 2(-2)^2 - (-2) - 1 = 2(4) + 2 - 1 = 8 + 2 - 1 = 9
\]

Since \(f(-2) = 9\), the point \((-2, 9)\) **is** on the graph of \(f\).

---

### (b) If \(x = 2\), what is \(f(x)\)? What point is on the graph of \(f\)?
Substitute \(x = 2\) into the function to find \(f(2)\):

\[
f(2) = 2(2)^2 - 2 - 1 = 2(4) - 2 - 1 = 8 - 2 - 1 = 5
\]

Thus, \(f(2) = 5\), and the point \((2, 5)\) is on the graph of \(f\).

---

### (c) If \(f(x) = -1\), what is \(x\)? What point(s) are on the graph of \(f\)?
To find \(x\) when \(f(x) = -1\), set \(f(x) = -1\) and solve for \(x\):

\[
2x^2 - x - 1 = -1
\]

Simplify the equation:

\[
2x^2 - x - 1 + 1 = 0 \quad \Rightarrow \quad 2x^2 - x = 0
\]

Factor the equation:

\[
x(2x - 1) = 0
\]

The solutions are:

\[
x = 0 \quad \text{or} \quad 2x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{2}
\]

Thus, \(x = 0\) and \(x = \frac{1}{2}\), so the points \((0, -1)\) and \(\left(\frac{1}{2}, -1\right)\) are on the graph of \(f\).

---

### (d) What is the domain of \(f\)?
Since \(f(x)\) is a quadratic function, it is defined for all real values of \(x\). Therefore, the domain of \(f\) is:

\[
\text{Domain of } f = (-\infty, \infty)
\]

---

### (e) List the \(x\)-intercept(s), if any, of the graph of \(f\).
The \(x\)-intercepts occur where \(f(x) = 0\). Set \(f(x) = 0\) and solve:

\[
2x^2 - x - 1 = 0
\]

Solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-1)}}{2(2)} = \frac{1 \pm \sqrt{1 + 8}}{4} = \frac{1 \pm \sqrt{9}}{4} = \frac{1 \pm 3}{4}
\]

So,

\[
x = \frac{1 + 3}{4} = 1 \quad \text{or} \quad x = \frac{1 - 3}{4} = -\frac{1}{2}
\]

Thus, the \(x\)-intercepts are \(x = 1\) and \(x = -\frac{1}{2}\). The points are \((1, 0)\) and \(\left(-\frac{1}{2}, 0\right)\).

---

### (f) List the \(y\)-intercept, if any, of the graph of \(f\).
The \(y\)-intercept occurs where \(x = 0\). Substituting \(x = 0\) into \(f(x)\):

\[
f(0) = 2(0)^2 - 0 - 1 = -1
\]

Thus, the \(y\)-intercept is \((0, -1)\).

---

### Summary of Answers:
- (a) Yes, the point \((-2, 9)\) is on the graph.
- (b) \(f(2) = 5\); the point is \((2, 5)\).
- (c) \(x = 0\) or \(x = \frac{1}{2}\); the points are \((0, -1)\) and \(\left(\frac{1}{2}, -1\right)\).
- (d) The domain of \(f\) is \((-\infty, \infty)\).
- (e) The \(x\)-intercepts are \((1, 0)\) and \(\left(-\frac{1}{2}, 0\right)\).
- (f) The \(y\)-intercept is \((0, -1)\).

---

Do you need any further details on any of the parts?

### Related questions:
1. What is the vertex of the quadratic function \(f(x)\)?
2. What is the axis of symmetry for the function \(f(x)\)?
3. How do you find the maximum or minimum value of a quadratic function?
4. What is the range of \(f(x) = 2x^2 - x - 1\)?
5. How do you graph the function \(f(x) = 2x^2 - x - 1\)?

### Tip: 
To solve quadratic equations efficiently, the quadratic formula is a powerful tool for finding roots when factoring is difficult.

Answer the questions about the following function.
f(x) = 2x^2 - x - 1
​(a)  Is the point (-2, 9) on the graph of​ f?
​(b)  If x​ = 2​, what is​ f(x)? What point is on the graph of​ f?
​(c)  If​ f(x) = -1​, what is​ x? What​ point(s) are on the graph of​ f?
​(d)  What is the domain of​ f?
​(e)  List the​ x-intercept(s), if​ any, of the graph of f.
​(f)  List the​ y-intercept, if​ any, of the graph of f.

Explore how to solve the quadratic function f(x) = 2x^2 - x - 1, including finding the domain, x-intercepts, y-intercepts, and points on the graph. Learn how to use the quadratic formula to find roots and interpret the graph of the function. Suitable for students in Grades 9-10.

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