Parabola Quiz 10.4

1) Simplify: 5a³(-2a)

3a^3

-10a^4

The given expression can be simplified by multiplying the coefficients and adding the exponents of the variable. In this case, the coefficient 5 and -2 multiply to give -10, and the variable "a" has an exponent of 3 in the first term and 1 in the second term. Therefore, the simplified expression is -10a^4.

Explanation

The given expression can be simplified by multiplying the coefficients and adding the exponents of the variable. In this case, the coefficient 5 and -2 multiply to give -10, and the variable "a" has an exponent of 3 in the first term and 1 in the second term. Therefore, the simplified expression is -10a^4.

2) For the graph of y = x^2 - 5, what are the coordinates of the vertex?

(0, -5)

(-5, 0)

The vertex of a parabola in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)). In this case, the equation is y = x^2 - 5, so a = 1, b = 0, and c = -5. Plugging these values into the formula, we get (-0/2(1), f(0/2(1))) = (0, f(0)) = (0, -5). Therefore, the coordinates of the vertex are (0, -5).

Explanation

The vertex of a parabola in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)). In this case, the equation is y = x^2 - 5, so a = 1, b = 0, and c = -5. Plugging these values into the formula, we get (-0/2(1), f(0/2(1))) = (0, f(0)) = (0, -5). Therefore, the coordinates of the vertex are (0, -5).

3) For the equation, y = (x + 1)(x - 3), what is the value of y when x = 1?

-1

-4

When x = 1, we can substitute this value into the equation y = (x + 1)(x - 3). By substituting x = 1, we get y = (1 + 1)(1 - 3) = (2)(-2) = -4. Therefore, the value of y when x = 1 is -4.

Explanation

When x = 1, we can substitute this value into the equation y = (x + 1)(x - 3). By substituting x = 1, we get y = (1 + 1)(1 - 3) = (2)(-2) = -4. Therefore, the value of y when x = 1 is -4.

4) What is the equation of the parabola that is produced by translating the graph of y = x² three units to the left?

Y = (x+3)^2

Y = x^2 - 3

The equation of the parabola that is produced by translating the graph of y = x^2 three units to the left is y = (x+3)^2. This is because when we translate a graph to the left, we subtract the amount of units we want to translate from the x-coordinate. In this case, we subtract 3 from x, resulting in (x+3). The rest of the equation remains the same, so we have y = (x+3)^2.

Explanation

The equation of the parabola that is produced by translating the graph of y = x^2 three units to the left is y = (x+3)^2. This is because when we translate a graph to the left, we subtract the amount of units we want to translate from the x-coordinate. In this case, we subtract 3 from x, resulting in (x+3). The rest of the equation remains the same, so we have y = (x+3)^2.

5) Describe the graph of y = -(x - 2)^2.

Translate y = -x^2 two units to the right

Translate y = -x^2 two units down

The given equation, y = -(x - 2)^2, is a translation of the graph of y = -x^2 two units to the right. This means that every point on the graph of y = -x^2 is shifted horizontally by two units to the right to obtain the graph of y = -(x - 2)^2. The negative sign in front of the equation indicates that the graph is reflected across the x-axis. The vertex of the graph is at (2, 0), which is two units to the right of the vertex of y = -x^2.

Explanation

The given equation, y = -(x - 2)^2, is a translation of the graph of y = -x^2 two units to the right. This means that every point on the graph of y = -x^2 is shifted horizontally by two units to the right to obtain the graph of y = -(x - 2)^2. The negative sign in front of the equation indicates that the graph is reflected across the x-axis. The vertex of the graph is at (2, 0), which is two units to the right of the vertex of y = -x^2.

6) Simplify: (-4x⁵y)⁴

256x^20y^4

256x^9y^4

To simplify the expression (-4x5y)4, we need to apply the exponent to each term inside the parentheses. The exponent 4 is distributed to both -4, x, 5, and y. Since (-4)^4 is equal to 256, x^5 raised to the power of 4 becomes x^20, and y^4 remains the same, the simplified expression is 256x^20y^4.

Explanation

To simplify the expression (-4x5y)4, we need to apply the exponent to each term inside the parentheses. The exponent 4 is distributed to both -4, x, 5, and y. Since (-4)^4 is equal to 256, x^5 raised to the power of 4 becomes x^20, and y^4 remains the same, the simplified expression is 256x^20y^4.

7) For the graph of y = x^2 + 7, what is the equation of the line of symmetry?

X = 7

X = 0

The equation of the line of symmetry for a parabola in the form y = ax^2 + bx + c is given by x = -b/2a. In this case, the equation y = x^2 + 7 is already in the standard form, so the coefficient b is 0. Therefore, the line of symmetry is x = -0/2(1) = 0.

Explanation

The equation of the line of symmetry for a parabola in the form y = ax^2 + bx + c is given by x = -b/2a. In this case, the equation y = x^2 + 7 is already in the standard form, so the coefficient b is 0. Therefore, the line of symmetry is x = -0/2(1) = 0.

8) What are the x-intercepts for the graph of y = (x+2)(2x-3)?

X = 2 and x = -3

X = -2 and x = 3

X = 3/2 and x = -2

The x-intercepts of a graph are the points where the graph intersects the x-axis. In order to find the x-intercepts, we set y equal to zero and solve for x. In the given equation y = (x+2)(2x-3), we set y = 0 and solve for x. By factoring the equation, we get (x+2)(2x-3) = 0. Setting each factor equal to zero, we find x = -2 and x = 3/2. Therefore, the x-intercepts for the graph of y = (x+2)(2x-3) are x = 3/2 and x = -2.

Explanation

The x-intercepts of a graph are the points where the graph intersects the x-axis. In order to find the x-intercepts, we set y equal to zero and solve for x. In the given equation y = (x+2)(2x-3), we set y = 0 and solve for x. By factoring the equation, we get (x+2)(2x-3) = 0. Setting each factor equal to zero, we find x = -2 and x = 3/2. Therefore, the x-intercepts for the graph of y = (x+2)(2x-3) are x = 3/2 and x = -2.