College Algebra Problem Bank 2 – Step-by-Step Study Guidance

Q1. Is the function $y = -x^2 - 4x - 8$ quadratic? If so, is its graph concave up or down, and is the vertex a maximum or minimum?

Background

Topic: Quadratic Functions and Their Graphs

This question tests your ability to recognize quadratic functions and analyze their graphs, including concavity and the nature of the vertex.

Key Terms and Formulas:

• Quadratic function: $y = ax^2 + bx + c$

• Concave up: $a > 0$ (parabola opens upward)

• Concave down: $a < 0$ (parabola opens downward)

• Vertex: The highest or lowest point on the graph

Step-by-Step Guidance

1. Compare the given function to the standard quadratic form $y = ax^2 + bx + c$ to determine if it is quadratic.

2. Identify the value of $a$ in the function.

3. Decide if the graph is concave up or down based on the sign of $a$.

4. Determine whether the vertex is a maximum or minimum point, depending on the concavity.

Try solving on your own before revealing the answer!

Q2. Is the function $y = 6x^3 - 7x^2$ quadratic? If so, is its graph concave up or down, and is the vertex a maximum or minimum?

Background

Topic: Identifying Quadratic Functions

This question checks your understanding of what makes a function quadratic and how to analyze its graph.

Key Terms and Formulas:

• Quadratic function: $y = ax^2 + bx + c$

• Degree of a function: The highest power of $x$

Step-by-Step Guidance

1. Identify the highest degree of $x$ in the function.

2. Recall that a quadratic function has degree 2.

3. Decide if the function is quadratic based on its degree.

4. If it is not quadratic, explain why.

Try solving on your own before revealing the answer!

Q3. Is the function $y = x^2 - 5x + 6$ quadratic? If so, is its graph concave up or down, and is the vertex a maximum or minimum?

Background

Topic: Quadratic Functions and Their Properties

This question asks you to identify a quadratic function and analyze its graph's shape and vertex.

Key Terms and Formulas:

• Quadratic function: $y = ax^2 + bx + c$

• Concavity: Determined by the sign of $a$

• Vertex: Maximum if concave down, minimum if concave up

Step-by-Step Guidance

1. Compare the function to the standard quadratic form.

2. Identify the coefficient $a$.

3. Determine the concavity based on $a$.

4. Decide if the vertex is a maximum or minimum.

Try solving on your own before revealing the answer!

Q4. Is the function $y = -3x^2 - 5x - 9$ quadratic? If so, is its graph concave up or down, and is the vertex a maximum or minimum?

Background

Topic: Quadratic Functions and Graph Analysis

This question tests your ability to recognize quadratic functions and analyze their graphs.

Key Terms and Formulas:

• Quadratic function: $y = ax^2 + bx + c$

• Concavity: $a > 0$ (up), $a < 0$ (down)

• Vertex: Maximum or minimum point

Step-by-Step Guidance

1. Check if the function matches the quadratic form.

2. Identify the value of $a$.

3. Determine the direction of the parabola (concave up or down).

4. State whether the vertex is a maximum or minimum.

Try solving on your own before revealing the answer!

Q5a. For $y = (x - 5)^2 + 1$, what are the coordinates of the vertex?

Background

Topic: Vertex Form of a Quadratic Function

This question asks you to identify the vertex of a parabola given in vertex form.

Key Terms and Formulas:

• Vertex form: $y = a(x - h)^2 + k$

• Vertex: $(h, k)$

Step-by-Step Guidance

1. Compare the given function to the vertex form $y = a(x - h)^2 + k$.

2. Identify the values of $h$ and $k$ from the equation.

3. Write the vertex as the point $(h, k)$.

Try solving on your own before revealing the answer!

Q5b. Graph the function $y = (x - 5)^2 + 1$ on the standard window.

Background

Topic: Graphing Quadratic Functions

This question asks you to sketch the graph of a quadratic function in vertex form.

Key Terms and Formulas:

• Vertex: $(h, k)$

• Axis of symmetry: $x = h$

• Standard window: Typically $x$$from$$to $10$y$ from $-10$ to $10$

Step-by-Step Guidance

1. Plot the vertex you found in part (a).

2. Draw the axis of symmetry through $x = h$.

3. Since $a = 1 > 0$, the parabola opens upward.

4. Plot a few additional points on either side of the vertex to help sketch the curve.

Try sketching the graph before checking the answer!

Q6a. For $y = -(x - 1)^2 - 2$, what are the coordinates of the vertex?

Background

Topic: Vertex Form of a Quadratic Function

This question asks you to find the vertex of a parabola given in vertex form.

Key Terms and Formulas:

• Vertex form: $y = a(x - h)^2 + k$

• Vertex: $(h, k)$

Step-by-Step Guidance

1. Compare the function to $y = a(x - h)^2 + k$.

2. Identify $h$ and $k$ (be careful with signs).

3. Write the vertex as $(h, k)$.

Try solving on your own before revealing the answer!

Q6b. Graph the function $y = -(x - 1)^2 - 2$ on the standard window.

Background

Topic: Graphing Quadratic Functions

This question asks you to sketch the graph of a downward-opening parabola in vertex form.

Key Terms and Formulas:

• Vertex: $(h, k)$

• Axis of symmetry: $x = h$

• Standard window: $x$$from$$to $10$y$ from $-10$ to $10$

Step-by-Step Guidance

1. Plot the vertex from part (a).

2. Draw the axis of symmetry at $x = h$.

3. Since $a = -1 < 0$, the parabola opens downward.

4. Plot a few points on either side of the vertex to help sketch the curve.

Try sketching the graph before checking the answer!