Function Basics Review Quiz

1) Identify the vertex and y-intercept of the graph of the function y = 3(x + 2)² - 5.

Vertex: (-2, -5)y-intercept: 7

Vertex: (2, 5)y-intercept: 12

Vertex: (2, -5)y-intercept: 7

Vertex: (-2, 5)y-intercept: 1

The vertex of a quadratic function in the form y = a(x - h)^2 + k is given by (h, k). In this case, the equation is y = 3(x + 2)^2 - 5, so the vertex is (-2, -5). The y-intercept is the value of y when x = 0. Plugging in x = 0 into the equation, we get y = 3(0 + 2)^2 - 5 = 3(4) - 5 = 12 - 5 = 7. Therefore, the y-intercept is 7.

Explanation

The vertex of a quadratic function in the form y = a(x - h)^2 + k is given by (h, k). In this case, the equation is y = 3(x + 2)^2 - 5, so the vertex is (-2, -5). The y-intercept is the value of y when x = 0. Plugging in x = 0 into the equation, we get y = 3(0 + 2)^2 - 5 = 3(4) - 5 = 12 - 5 = 7. Therefore, the y-intercept is 7.

2) Which quadratic function does the graph represent?

F(x) = -x2 + 6x + 7

F(x) = x2 + 6x - 7

F(x) = -x2 + 6x - 7

F(x) = -x2 - 6x - 7

The given quadratic function is f(x) = -x^2 + 6x - 7. This can be determined by analyzing the coefficients of the quadratic equation. The quadratic term has a negative coefficient, indicating that the graph opens downwards. The linear term has a positive coefficient, indicating that the graph has a positive slope. The constant term is negative, indicating that the graph intersects the y-axis below the origin. Therefore, the correct answer is f(x) = -x^2 + 6x - 7.

Explanation

The given quadratic function is f(x) = -x^2 + 6x - 7. This can be determined by analyzing the coefficients of the quadratic equation. The quadratic term has a negative coefficient, indicating that the graph opens downwards. The linear term has a positive coefficient, indicating that the graph has a positive slope. The constant term is negative, indicating that the graph intersects the y-axis below the origin. Therefore, the correct answer is f(x) = -x^2 + 6x - 7.

3) Write the equation of the parabola in vertex form. vertex: (-2, 4), point: (2, 84)

Y = 5(x - 2)2 + 4

Y = 84(x + 2)2 - 4

Y = 5(x + 2)2+4

Y = 2(x - 2)2 + 4

The equation of a parabola in vertex form is given by y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola. In this case, the vertex is (-2, 4), so the equation should be of the form y = a(x + 2)^2 + 4. Since the given equation y = 5(x + 2)^2 + 4 matches this form, it is the correct answer.

Explanation

The equation of a parabola in vertex form is given by y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola. In this case, the vertex is (-2, 4), so the equation should be of the form y = a(x + 2)^2 + 4. Since the given equation y = 5(x + 2)^2 + 4 matches this form, it is the correct answer.

4) Use vertex form to write the equation of the parabola.

Y = 2(x - 3)2 - 2

Y = 2(x + 3)2 - 2

Y = 2(x + 3)2 + 2

Y = (x - 3)2 - 2

The given equation is in the vertex form of a parabola, which is y = a(x - h)^2 + k. In this equation, the vertex of the parabola is represented by the point (h, k).

Comparing the given equation y = 2(x - 3)^2 - 2 with the vertex form, we can see that the vertex of the parabola is at the point (3, -2). The coefficient "2" in front of the (x - 3)^2 term indicates that the parabola is stretched vertically. Since the coefficient is positive, the parabola opens upward. Therefore, the equation y = 2(x - 3)^2 - 2 represents a parabola with its vertex at (3, -2), opening upward, and stretched vertically.

Explanation

The given equation is in the vertex form of a parabola, which is y = a(x - h)^2 + k. In this equation, the vertex of the parabola is represented by the point (h, k).

Comparing the given equation y = 2(x - 3)^2 - 2 with the vertex form, we can see that the vertex of the parabola is at the point (3, -2). The coefficient "2" in front of the (x - 3)^2 term indicates that the parabola is stretched vertically. Since the coefficient is positive, the parabola opens upward. Therefore, the equation y = 2(x - 3)^2 - 2 represents a parabola with its vertex at (3, -2), opening upward, and stretched vertically.

5) The parent function f(x) = x² is reflected across the x-axis, vertically stretched by a factor of 3, and translated right 7 units to create g. Use the description to write the quadratic function in vertex form.

G(x) = 7(x + 3)2

G(x) = -3(x - 7)2

G(x) = -3(x + 7)2

G(x) = 3(x - 7)2

The parent function f(x) = x^2 is reflected across the x-axis, vertically stretched by a factor of 3, and translated right 7 units to create g. The equation g(x) = -3(x - 7)^2 represents this transformation. The negative sign indicates the reflection across the x-axis, the coefficient -3 represents the vertical stretch, and the (x - 7)^2 term represents the translation right 7 units. Thus, the correct answer is g(x) = -3(x - 7)^2.

Explanation

The parent function f(x) = x^2 is reflected across the x-axis, vertically stretched by a factor of 3, and translated right 7 units to create g. The equation g(x) = -3(x - 7)^2 represents this transformation. The negative sign indicates the reflection across the x-axis, the coefficient -3 represents the vertical stretch, and the (x - 7)^2 term represents the translation right 7 units. Thus, the correct answer is g(x) = -3(x - 7)^2.