Integrated math2: homework2.5 graphing quadratics in standard form

Answer 1

Graphing quadratics in standard form involves the quadratic equation expressed as:

\[ y = ax^2 + bx + c \]

where:

• \( a \) determines the direction and width of the parabola.
• \( b \) affects the position of the vertex along the x-axis.
• \( c \) is the y-intercept (the point where the graph crosses the y-axis).

Steps to Graph a Quadratic in Standard Form:

1. Identify \( a \), \( b\), and \( c \): Determine the coefficients from the standard form of the equation.

2. Find the Vertex:

   • The x-coordinate of the vertex can be found using the formula: \[ x = -\frac{b}{2a} \]
   • Substitute this \( x \) value back into the quadratic equation to find the y-coordinate of the vertex.

3. Determine the Direction:

   • If \( a > 0 \), the parabola opens upwards.
   • If \( a < 0 \), the parabola opens downwards.

4. Find the Y-Intercept:

   • The y-intercept occurs at \( x = 0 \), which gives the point \( (0, c) \).

5. Find X-Intercepts (if applicable):

   • To find the x-intercepts, set \( y = 0 \) and solve the equation \( ax^2 + bx + c = 0 \).
   • You can use factoring, completing the square, or the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

6. Graph the Quadratic:

   • Plot the vertex, y-intercept, and x-intercepts on the coordinate plane.
   • Draw a smooth curve through these points to form the parabola.

Example Problem:

Graph the following quadratic function:

\[ y = 2x^2 - 4x + 1 \]

1. Identify coefficients: \( a = 2, b = -4, c = 1 \)

2. Find the vertex: \[ x = -\frac{-4}{2 \cdot 2} = \frac{4}{4} = 1 \] Now find \( y \): \[ y = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1 \] Vertex: \( (1, -1) \)

3. Determine direction: Since \( a = 2 > 0 \), the parabola opens upwards.

4. Find the y-intercept: \[ (0, c) = (0, 1) \]

5. Find x-intercepts: \[ 0 = 2x^2 - 4x + 1 \] Using the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} = \frac{4 \pm \sqrt{16 - 8}}{4} = \frac{4 \pm \sqrt{8}}{4} = \frac{4 \pm 2\sqrt{2}}{4} = 1 \pm \frac{\sqrt{2}}{2} \]

6. Graph the quadratic:

   • Plot the points: vertex \( (1, -1) \), y-intercept \( (0, 1) \), and x-intercepts.
   • Draw the parabola opening upwards through these points.

This process will allow you to graph any quadratic equation in standard form! If you have specific equations from your homework, feel free to provide them, and I can help you graph those as well.