91Ó°ÊÓ

Quadratic functions graph into a shape known as a parabola. Parabolas have a unique, symmetric, U-like shape. The direction of this curve depends on the sign of the coefficient 'a' in the quadratic expression. For the function \[ y = \frac{1}{3}x^2 - 4x + 15 \]- Since the coefficient \( a = \frac{1}{3} \) is positive, the parabola opens upwards.- A parabola is symmetric about a vertical line called the axis of symmetry, which passes through the vertex of the parabola.When graphing a quadratic function:- Start by identifying the vertex, the highest or lowest point depending on the graph's orientation.- The y-intercept provides an initial point on the graph.- A parabola's curvature is smooth and continuous, without sharp angles.

The vertex of a parabola is an important feature. It represents either a maximum or minimum point on the graph. To find the vertex from the standard form equation \[ y = ax^2 + bx + c \]you use the formula \[ x = -\frac{b}{2a} \].Substituting for the given function:- The values are \( a = \frac{1}{3} \), \( b = -4 \), and \( c = 15 \).- Calculate \( x \) coordinate: \(-\frac{-4}{2 \times \frac{1}{3}} = 6\).- For the y-coordinate, substitute \( x = 6 \) back into the function: - \( y = \frac{1}{3}(6)^2 - 4(6) + 15 = 3 \).Thus, the vertex of the parabola is \((6, 3)\). This point is vital for determining the graph's symmetry and shape.

Intercepts are points where the graph crosses the axes. Finding these points helps us understand the position and shape of the parabola.### Y-Intercept- The y-intercept is where the graph crosses the y-axis, so set \( x = 0 \) in the equation: \[ y = \frac{1}{3}(0)^2 - 4(0) + 15 = 15 \].- Thus, the y-intercept is at \((0, 15)\).### X-Intercepts- These are points where the graph crosses the x-axis, found by setting \( y = 0 \) and solving the equation: - Rearrange: \( x^2 - 12x + 45 = 0 \). - The discriminant \( b^2 - 4ac \) is negative, specifically \( \sqrt{-36} \), indicating no real x-intercepts.- Therefore, the parabola does not cross the x-axis. This occurs when a parabola lies entirely above or below the x-axis, confirming no real roots.

The quadratic formula is a crucial tool for finding the roots of any quadratic equation. The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].- For our equation \( x^2 - 12x + 45 = 0 \), we have \( a = 1 \), \( b = -12 \), and \( c = 45 \). - Plug these values into the formula: \[ x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(45)}}{2(1)} \]. - Evaluate the discriminant: \( 144 - 180 = -36 \).This negative discriminant means the square root of a negative number, indicating no real solutions for x-intercepts. The knowledge of the quadratic formula thus helps us identify when solutions to a quadratic equation exist in the real number system.