Question

Please help!


Algebra 3


thanks

Answer 1

The y-intercept is (0, -3).

The axis of symmetry is x=3/4.

The vertex is (3/4, -21/4).

Explanation:

We are given the following quadratic function.

`f(x)=4x^2-6x-3`

And we are asked to determine the following:

• y-intercept(s)
• Axis of symmetry
• Vertex

`\hrulefill`

To find the y-intercept, axis of symmetry, and vertex of a quadratic function, you can follow these steps:

(1) - Identify the quadratic function: Determine the quadratic function for which you want to find the y-intercept, axis of symmetry, and vertex. It is usually given as an equation or described in a problem.

(2) - Y-intercept: To find the y-intercept, substitute x = 0 into the quadratic function and evaluate the expression. The resulting value represents the y-coordinate of the point where the graph intersects the y-axis.

(3) - Axis of symmetry: The axis of symmetry is a vertical line that passes through the vertex of the quadratic function. To find the axis of symmetry, you can use one of the following methods:

• If the quadratic function is in vertex form, f(x) = a(x - h)² + k, then the axis of symmetry is given by the equation x = h, where (h, k) represents the vertex of the parabola.

• If the quadratic function is in standard form, f(x) = ax² + bx + c, you can use the formula x = -b / (2a) to find the x-coordinate of the vertex.

(4) - Vertex: The vertex of a quadratic function represents the highest or lowest point on the graph (the maximum or minimum point). To find the vertex, you can use one of the following methods:

• If the quadratic function is in vertex form, the vertex is directly given as (h, k).

• If the quadratic function is in standard form, you can substitute the x-coordinate obtained from the axis of symmetry into the function to find the corresponding y-coordinate. The vertex is then represented by the point (x, y).

`\hrulefill`

Step (1) -

`f(x)=4x^2-6x-3`

Step (2) -

`\Longrightarrow f(0)=4(0)^2-6(0)-3\\\\\\\therefore \boxed{f(0)=-3}`

Thus, the y-intercept is (0, -3).

Step (3) -

The given function is in standard form. Thus, we can use the following formula:

`x=(-b)/(2a); \ \text{In our case:} \ b=-6 \ \text{and} \ a=4\\ \\\\\Longrightarrow x=(-(-6))/(2(4))\\\\\\\Longrightarrow x=(6)/(8)\\\\\\\therefore \boxed{x=(3)/(4) }`

Thus, the axis of symmetry is found.

Step (4) -

Recall that we were given a function in standard form and in step 3 we found that x=3/4.

`\Longrightarrow f((3)/(4) )=4((3)/(4))^2-6((3)/(4) )-3\\\\\\\Longrightarrow f((3)/(4) )=(9)/(4)-(9)/(2)-3\\\\\\\therefore \boxed{ f((3)/(4) )= -(21)/(4) }`

Thus, the vertex is (3/4, -21/4).