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3.7 – Graphs of Quadratic Functions
- Determine the number of times that a given function crosses the x-axis.
- Find the vertex and x-intercepts of the graph of a quadratic function.
- Use the quadratic formula to find the vertex of a given function.
- Determine if a discriminant is positive, negative, or zero when given the graph of the function.
- Match a graph to its quadratic equation.
- Average Rate of Change – A quotient that tells, on average, how some quantity changes over time.
- The ratio between the change in the dependent variable and the change in the independent variable.
- Instantaneous Slope – The slope of a line that is tangent to a curve at a point.
- It may be estimated by calculating the average slope between the point and a nearby point on the curve.
- The closer the nearby point, the better the estimation will be.
- In nonlinear functions, the average rate of change and the instantaneous slope are NOT the same thing.
- Vertex – Marks four features of a parabola:
- Its sharpest turn
- Its center
- Its line of symmetry
- The halfway point between its two x-intercepts
- Roots of a Polynomial
- The x-intercepts
- The places where the graph crosses, or touches, the x-axis
- The x-values that make y = 0
- Roots are also called Zeros (where the parabola crosses the x-axis)
- To find the zeros of a parabola, set the y equal to 0 and solve for x.
- Use any of the methods learned in this chapter to solve for x.
- To find the average of 2 values, add the values together and divide by 2.
- Ex. 4 and 14
- Add 4 + 14: 18
- Divide by 2: 9
- The average of 4 and 14 is 9.
|Quadratics’ Graphs are Parabolas
- When quadratic equations are graphed, they form parabolas.
- The vertex of a parabola is the point at which the direction of the function changes.
- The vertex of a parabola is also either the minimum or the maximum of the function, depending on whether the leading coefficient is positive or negative.
- Minimum: parabola opens upward
- Maximum: parabola opens downward
|Finding the Vertex of a Parabola
- Option 1: Finding the Vertex by Factoring & finding Roots
- 1st: find the x-intercepts (roots)
- 2nd: factor
- 3rd: set equal to zero & solve
- 4th: list the roots as coordinates (x,0) and (x,0)
- 5th: find the average of the x-values
- 6th: substitute the average into the original equation and find the y-value
- 1st & 2nd: factors into (x – 1)(x – 5) = 0
- 3rd: x – 1 = 0 results in x = 1
- 3rd: x – 5 = 0 results in x = 5
- 4th: The roots are (1,0) and (5,0) since we know the roots are on the x-axis (where the y-value is equal to zero)
- 5th: find the average:
- 6th: substitute the x-value for the vertex into the original equation to find the y-value:
- Remember the signs of the equation and the signs of the binomial factors (x – #)(x + #) can help find the roots
- Option 2: Finding the Vertex of a parabola with the Quadratic Formula
- Formula for finding the x-value of the vertex:
* Imaginary numbers ARE complex roots
- Example 1:
- a: 1
- b: -6
- c: 5
- Solve for the x-value: 3
- Substitute 3 for x and solve for y (see step 6 above)
|Finding the Equation of a Parabola
- Parabolas are symmetrical curves, meaning that each side of the curve is an exact reflection of the other side.
- The line of symmetry (also called the axis of symmetry) that divides the parabola into two mirror images runs directly through its vertex, as shown in the image below.
- To find the equation of a parabola, set up the binomial factors as the opposite values of the roots (zeros)
- When a is positive, the root is (x – a).
- When a is negative, the root is (x + a).
- Ex. Roots: x = 6 and x = 2 will be the factors: (x – 6) and (x – 2). Use FOIL to find the equation:
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