Question

UNDERSTANDING GRAPHS Sketch the graphs of the three functions in the same coordinate plane. Describe how the three graphs are related. a. \(y=x^{2}+x+1\) \(y=\frac{1}{2} x^{2}+x+1\) \(y=2 x^{2}+x+1\) b. \(y=x^{2}-1 x+1\) \(y=x^{2}-5 x+1\) \(y=x^{2}-10 x+1\) c. \(y=x^{2}-x+1\) \(y=x^{2}-x+3\) \(y=x^{2}-x-2\)

Short answer

For each case: In a, all functions have the same vertex, but differing steepness with \(y=2x^2 + x + 1\) being the narrowest. For b, all parabolas have their vertex moving further to the right as the magnitude of \(b\), the coefficient of \(x\), increases. In c, an increase in the constant term results in the graph being higher on the coordinate plane.

Step-by-step solution

Identify the vertices and shape of the graphs

Firstly, identify the coordinates of the vertex \((-b/2a, -D/4a)\), where \(a\), \(b\), \(c\) are the coefficients in the general form of a quadratic function \(y=ax^2+bx+c\), and \(D=b^2-4ac\) stands for the discriminant. Then, check the sign of \(a\): if \(a>0\), the graph opens upwards (a minimum point), if \(a<0\), it opens downwards (a maximum point). Observe how the shapes vary with the coefficient \(a\).

Observe the changes when \(a\) varies

Next, compare the three functions in each case, focusing on the coefficient of \(x^2\) (which is \(a\) in \(y = ax^2 + bx + c\)). In case a, the coefficients are 1, 1/2, and 2 respectively. Since the coefficient changes while the others remain the same, the function will have the same vertex, but different steepness. For example, \(y=2x^2 + x + 1\) will be narrower than the other two.

Observe the shift when \(b\) varies

In case b, it is the coefficient of \(x\) that varies while the others stay the same. The change in this coefficient results in a horizontal shift of the graph. As the magnitude of the coefficient increases, the vertex moves further to the right.

Observe the shift when the constant term varies

Finally, for case c, the only difference is in the constant term, affecting the vertical position of the graph. A function with a greater constant term will be simply lifted up compared to the others. The greater the constant term, the higher the graph.

Key concepts

Vertex of a Quadratic Function

The vertex of a quadratic function is a crucial point that determines the function’s peak or trough. When graphing a quadratic function in the form of y = ax^2 + bx + c, finding the vertex gives us a clear starting point. Calculate the vertex's x-coordinate using the formula x = -b/(2a). Once you have the x-coordinate, find the y-coordinate by substituting this value back into the original equation.

For the given functions in our exercise, we would use this formula to understand the shape and location of each parabola on the coordinate plane. For instance, the vertex for the function y=x^2+x+1 is found by first calculating x as -1/(2*1), which simplifies to -1/2, and then finding the y-coordinate by inserting -1/2 back into the function. The significance of the vertex is not only in graphing but also in understanding the maximum or minimum values that the function can reach.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation, represented by D or sometimes Delta, tells us about the nature of the roots of the equation ax^2 + bx + c = 0. It is calculated using the formula D = b^2 - 4ac. Here's how it aides our understanding:

• If D > 0, the equation has two distinct real roots.
• If D = 0, there is exactly one real root, or the roots are repeated.
• If D < 0, the roots are complex and not real.

For graphing, the discriminant is also useful. If D >= 0, the graph of the quadratic function will touch or cross the x-axis, while D < 0 indicates that the graph will not intersect the x-axis. In our exercise, the discriminant would affect where the vertex of our parabola is located in relation to the x-axis and allows for prediction of the graph's intersection points with the axis without actually solving the function.

Coefficient Effects on Parabolas

The coefficients in a quadratic function, y = ax^2 + bx + c, have distinct effects on the graph of a parabola. The coefficient a affects the width and direction of the parabola; a higher absolute value of a makes the parabola narrower, and if a is negative, the parabola opens downwards.

As demonstrated in our exercise, changing the value of a results in different steepness of the parabolas without changing the location of their vertices. This is because only the vertical stretch is affected, not the horizontal position.

Impact of Coefficient b

The coefficient b influences the horizontal shift of the parabola. As the value of b changes, the vertex of the parabola moves along the x-axis. A higher magnitude of b means a more significant shift.

Role of the Constant Term c

The constant term c affects the vertical position. It determines where the parabola crosses the y-axis, which is also the y-coordinate of the vertex when b = 0. Altering c moves the parabola up or down without changing its shape. The carefully crafted graphical shifts observed in this exercise reveal how each coefficient independently influences the graph of a quadratic function.