In this problem we will explore the effect that changing the...

In this problem we will explore the effect that changing the coefficient of $x$ has on the graph of the parabola $y=3 x^2+b x-2$. a. First consider positive values of $b$. Use the computer to graph $y=3 x^2+x-2$, $y=3 x^2+2 x-2$, and $y=3 x^2+5 x-2$ on the same set of axes. How does changing the value of $b$ affect the vertex of $y=3 x^2+b x-2$ ? Feel free to try additional positive values of $b$ to convince yourself that your answer is correct. b. Now graph $y=3 x^2+b x-2$ for $b=-1,-2$, and -4 . When the coefficient $b$ is negative, how does changing its value affect the parabola? c. Summarize the effect that the value of $b$ has on the position of the vertex of the parabola. Make sure that your answer is valid for all values of $b$, positive, negative or zero. d. Notice that all the graphs have the same $y$-intercept. Give a simple explanation for this phenomenon.

In this problem we will explore the effect that changing the coefficient of $x$ has on the graph of the parabola $y=3 x^2+b x-2$.
a. First consider positive values of $b$. Use the computer to graph $y=3 x^2+x-2$, $y=3 x^2+2 x-2$, and $y=3 x^2+5 x-2$ on the same set of axes. How does changing the value of $b$ affect the vertex of $y=3 x^2+b x-2$ ? Feel free to try additional positive values of $b$ to convince yourself that your answer is correct.
b. Now graph $y=3 x^2+b x-2$ for $b=-1,-2$, and -4 . When the coefficient $b$ is negative, how does changing its value affect the parabola?
c. Summarize the effect that the value of $b$ has on the position of the vertex of the parabola. Make sure that your answer is valid for all values of $b$, positive, negative or zero.
d. Notice that all the graphs have the same $y$-intercept. Give a simple explanation for this phenomenon.

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial function in the form ax² + bx + c. Its graph is a parabola, which can open upward or downward depending on the sign of the coefficient a. Understanding quadratic functions involves examining how each coefficient (a, b, and c) influences the graph's shape and position in the coordinate plane.

Vertex of a Parabola

The vertex of a parabola is the point where the function reaches its maximum or minimum value. For a quadratic function in standard form (ax² + bx + c), the x-coordinate of the vertex is given by -b/(2a). This formula shows that changes in the coefficient b directly affect the horizontal position of the vertex, while the y-coordinate is determined by substituting this x-value back into the equation.

Effect of the Linear Coefficient (b)

The coefficient b in a quadratic function influences the location of the vertex along the x-axis. Since the vertex's x-coordinate is calculated as -b/(2a), increasing or decreasing the value of b results in a horizontal shift of the vertex. This shift affects the symmetry of the parabola without altering the overall shape determined by the coefficient a.

Y-Intercept Invariance

The y-intercept of a quadratic function is the point where the graph crosses the y-axis, which is found by evaluating the function at x = 0. Since the y-intercept is given by the constant term c (when x = 0, ax² and bx terms vanish), changing the coefficient b does not affect the y-intercept. This demonstrates that while b influences the vertex's horizontal position, it does not alter the point at which the graph meets the y-axis.

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