How do you find the vertex and the intercepts for #f(x) = 4x^2 - 16x +21#?

Question

Eva Adamson · Accepted Answer

Vertex (2, 5)

#f(x) = 4x^2 - 16x + 21# x-coordinate of vertex: #x = -b/(2a) = 16/8 = 2.# y-coordinate of vertex: #y(2) = 16 - 32 + 21 = 5# Vertex (2, 5) To find y-intercept, make x = 0 --> y = 21 To find x-intercepts, make y = 0 and solve the quadratic equation: #y = 4x^2 - 16x + 21 = 0# #D = b^2 - 4ac = 256 - 336 = - 80.# Since D < 0, there are no real roots, no x-intercepts. Since a > 0 the parabola opens upward and it is completely above the x-axis.

Hazel Anglin · Answer

undefined To find the vertex of the quadratic function $ f(x) = 4x^2 - 16x + 21 $, you can use the formula $ x = -\frac{b}{2a} $, where $ a = 4 $ and $ b = -16 $. 
To find the y-coordinate of the vertex, substitute the x-coordinate you found into the original function. 
To find the x-intercepts, set $ f(x) = 0 $ and solve for x. 
To find the y-intercept, evaluate $ f(x) $ when $ x = 0 $.

The vertex coordinates are $ (\frac{b}{2a}, f(\frac{b}{2a})) $. 
The x-intercepts are the solutions to $ f(x) = 0 $. 
The y-intercept is $ f(0) $.

Eli Assouline · Answer

undefined To find the vertex and the intercepts for the quadratic function $ f(x) = 4x^2 - 16x + 21 $, follow these steps:

1. **Vertex**:
   The vertex of a quadratic function in the form $ f(x) = ax^2 + bx + c $ is given by the formula:
   $$ x_{\text{vertex}} = -\frac{b}{2a} $$
   Once you find $ x_{\text{vertex}} $, substitute it into the function to find the corresponding $ y $-coordinate, which gives the vertex as $ (x_{\text{vertex}}, f(x_{\text{vertex}})) $.

2. **Intercepts**:
   - **$ x $-intercepts (or roots)**: These are the points where the graph intersects the $ x $-axis. To find them, set $ f(x) = 0 $ and solve for $ x $. These are also known as the solutions to the quadratic equation $ 4x^2 - 16x + 21 = 0 $.
   - **$ y $-intercept**: This is the point where the graph intersects the $ y $-axis. To find it, evaluate the function at $ x = 0 $, giving the point $ (0, f(0)) $.

By following these steps, you can find the vertex and intercepts for the given quadratic function $ f(x) = 4x^2 - 16x + 21 $. If you need further clarification on any step, feel free to ask.