- The parabola opens upward
- The x-intercepts at (-4, 0) and (1, 0)
- The y-intercept is the point (0, -4)
- The axis of symmetry is x = - ³/₂
- The minimum value is -6¹/₄
- The vertex is (- ³/₂, - 6¹/₄)
Further explanation
The quadratic function is described by the standard equation [tex]\boxed{ \ f(x) = ax^2 + bx + c \ }.[/tex]
We have the quadratic function [tex]\boxed{ \ f(x) = (x - 1)(x + 4) \ }.[/tex]
Let us configure it to obtain a standard equation.
[tex]\boxed{f(x) = (x - 1)(x + 4)}[/tex]
[tex]\boxed{f(x) = x^2 + 4x - x - 4}[/tex]
Hence, we get [tex]\boxed{\boxed{ \ f(x) = x^2 + 3x - 4 \ }}.[/tex]
- We identify the coefficients a, b, and c. For this equation, [tex]\boxed{ \ a = 1, b = 3, and \ c = -4 \ }[/tex]
- The parabola opens upward because a > 0, resulting in a vertex that is a minimum.
- The y-intercept of the quadratic function f(x) = x² + 3x - 4 is (0, c), i.e., the point [tex]\boxed{ \ (0, -4) \ }.[/tex]
- From [tex]\boxed{ \ f (x) = (x - 1)(x + 4) \ }[/tex] we get the x-intercepts at [tex]\boxed{ \ (-4, 0) \ and \ (1, 0) \ }[/tex]
- The axis of symmetry is [tex]\boxed{ \ x = h = -\frac{b}{2a} \ }[/tex], i.e., [tex]\boxed{ \ x = h = -\frac{3}{2(1)} \rightarrow h = -\frac{3}{2} \ }[/tex]
- The minimum value is [tex]\boxed{ \ k = -\frac{25}{4} = -6\frac{1}{4} \ }[/tex]
- The vertex is [tex]\boxed{ \ (h, k) \ },[/tex] where [tex]\boxed{ \ k = f(h) \ }[/tex] or [tex]\boxed{ \ k = \frac{b^2 - 4ac}{-4a} \ }[/tex]
Finding the minimum value is as follows:
- [tex]\boxed{ \ k = f (- \frac{3}{2}) = (- \frac{3}{2})^2 + 3(- \frac{3}{2}) - 4 = -\frac{25}{4} = -6\frac{1}{4} \ }, or[/tex]
- [tex]\boxed{ \ k = \frac{3^2 - 4(1)(-4)}{-4(1)} = -\frac{25}{4} = -6\frac{1}{4} \ }[/tex]
Notes:
- The graph of a quadratic function is called a parabola.
- When a > 0, the parabola opens upward, resulting in a vertex that is a minimum.
- When a < 0, the parabola opens downward, resulting in a vertex that is a maximum.
- The value c is the y-intercept of the graph, because a y-intercept is a point on the graph where x is zero. In other words, the graph passes through the point [tex]\boxed{ \ (0, c) \ }.[/tex]
- From [tex]\boxed{ \ f (x) = (x - x_1)(x - x_2) \ }[/tex] we get x-intercepts at [tex]\boxed{ \ (x_1, 0) \ and \ (x_2, 0) \ }.[/tex]
- An x-intercept represents a point on the graph where y is zero.
- The axis of symmetry represent the line that passes through the vertex of parabola with equation [tex]\boxed{ \ x = h = -\frac{b}{2a} \ }[/tex]
- The vertex is [tex]\boxed{ \ (h, k) \ },[/tex] where [tex]\boxed{ \ k = f(h) \ }[/tex] or [tex]\boxed{ \ k = \frac{b^2 - 4ac}{-4a} \ }[/tex]
Learn more
- A line that is not parallel to either the x-axis or the y-axis https://brainly.com/question/4691222
- Finding the y-intercept of the quadratic function f(x) = (x – 6)(x – 2) https://brainly.com/question/1332667
- The midpoint https://brainly.com/question/3269852
Keywords: which is the graph of f(x) = (x - 1)(x + 4), the x-intercept, quadratic function, a standard equation, the y-intercept, the axis of symmetry, the vertex, parabola, upward, downward
