Y=4x²-5x+1
To prepare for exams, it's important to be familiar with various aspects of a given equation. In this case, the equation is y = 4x² - 5x + 1.
Graph: You can plot the graph of this equation to get a visual representation of the equation's behavior. To do this, select a range of x-values, substitute them into the equation to find the corresponding y-values, and plot the points. Connect the plotted points to obtain the graph.
Vertex form: Rewrite the equation in vertex form to identify the vertex, which is the turning point of the quadratic function. The vertex form is given by y = a(x - h)² + k, where (h, k) represents the coordinates of the vertex. To convert the equation to vertex form, you can complete the square.
Starting with y = 4x² - 5x + 1: y = 4(x² - (5/4)x) + 1 y = 4(x² - (5/4)x + (5/8)² - (5/8)²) + 1 y = 4[(x - (5/8))² - (25/64)] + 1 y = 4(x - (5/8))² - (100/64) + 1 y = 4(x - (5/8))² - (100/64) + (64/64) y = 4(x - (5/8))² - (36/64) y = 4(x - (5/8))² - (9/16)
So, the equation in vertex form is y = 4(x - (5/8))² - (9/16). The vertex is at (5/8, -9/16).
- Discriminant: The discriminant of a quadratic equation provides information about the nature of the roots. It is calculated as b² - 4ac. In this equation, a = 4, b = -5, and c = 1. Calculate the discriminant to determine how many real roots the equation has:
Discriminant = (-5)² - 4(4)(1) = 25 - 16 = 9
Since the discriminant is positive (greater than zero), the equation has two distinct real roots.
- Axis of symmetry: The axis of symmetry is a vertical line through the vertex that divides the parabola into two symmetrical halves. The equation for the axis of symmetry can be found using the formula x = -b/2a. In this case, b = -5 and a = 4.
Axis of symmetry: x = -(-5)/(2*4) = 5/8
Therefore, the equation for the axis of symmetry is x = 5/8.
By familiarizing yourself with these aspects of the equation y = 4x² - 5x + 1, you will be better prepared for your exams. Good luck!
