Question

Given: f(x) = x^2 - 7x + 6

A. Find the vertex.
B. Find the y-intercept, compute f(0).
C. Find the x-intercepts, ax^2 + bx + c = 0
D.All of the above

Answer 1

D. All of the above to find the x-intercepts, we solve
`\(x^2 - 7x + 6 = 0\)` by factoring or using the quadratic formula, obtaining the x-intercepts at
`\(x = 1\) and \(x = 6\).`

Explanation:

The given quadratic function is \(f(x) = x² - 7x + 6\). To find the vertex, we utilize the formula
`\(x = -(b)/(2a)\)` for the x-coordinate and then substitute this value into the function to get the y-coordinate of the vertex. For the y-intercept, we compute \(f(0)\), which gives us the value of the function atx = 0. Finally, to find the x-intercepts, we set \(f(x) = 0\) and solve the resulting quadratic equation
`\(x^2 - 7x + 6 = 0\)` using methods like factoring, quadratic formula, or completing the square.

The vertex of the quadratic function is found by employing the formula for the x-coordinate of the vertex, which is
`\(-(-7)/(2 * 1)\),`resulting in \(x = \frac{7}{2}\). Substituting this x-value into the function gives us
`\(f\left((7)/(2)\right) = \left((7)/(2)\right)^2 - 7 * (7)/(2) + 6 = (1)/(4)\),`therefore the vertex is at \
`(\left((7)/(2), (1)/(4)\right)\).` For the y-intercept,
`\(f(0)\)` equals
`\(0^2 - 7 * 0 + 6 = 6\).` This implies the y-intercept is at \((0, 6)\). Finally, to find the x-intercepts, we solve
`\(x^2 - 7x + 6 = 0\)` by factoring or using the quadratic formula, obtaining the x-intercepts at
`\(x = 1\) and \(x = 6\).`

Answer 2

• A. The vertex of the function f(x) = x^2 - 7x + 6 can be found using the formula x = -b / (2a) for the x-coordinate of the vertex. Once you find x, substitute it into the equation to get the y-coordinate.
• B. To find the y-intercept, substitute x = 0 into the function f(x) = x^2 - 7x + 6 and compute f(0). This will give you the value of the function at x = 0, which is the y-intercept.
• C. To find the x-intercepts, set f(x) = x^2 - 7x + 6 equal to 0 and solve for x using the quadratic formula or factoring.

Step-by-step explanation:

1. A. The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b / (2a). For the given function f(x) = x^2 - 7x + 6, the x-coordinate of the vertex is x = -(-7) / (2 * 1) = 7 / 2. To find the y-coordinate, substitute x = 7 / 2 into the function: f(7 / 2) = (7 / 2)^2 - 7 * (7 / 2) + 6 = 49 / 4 - 49 / 2 + 6 = 1 / 4. Therefore, the vertex is at (7 / 2, 1 / 4).
2. B. To determine the y-intercept, substitute x = 0 into the function: f(0) = (0)^2 - 7 * 0 + 6 = 0 - 0 + 6 = 6. Hence, the y-intercept is 6, and the point is (0, 6).
3. C. For the x-intercepts, set f(x) = x^2 - 7x + 6 equal to 0: x^2 - 7x + 6 = 0. Factorizing or using the quadratic formula x = [7 ± √(7^2 - 4 * 1 * 6)] / (2 * 1), x = [7 ± √(49 - 24)] / 2, x = [7 ± √25] / 2, x = (7 ± 5) / 2. So, the x-intercepts are x = 2 and x = 5, making the points (2, 0) and (5, 0) respectively.