Step-by-Step Examples: Finding Intercepts of Quadratic Functions

📚 Quick Study Guide

🔍 What are Intercepts? Intercepts are the points where a quadratic function intersects the x-axis (x-intercepts) and the y-axis (y-intercept). 📈 Y-Intercept: To find the y-intercept, set $x = 0$ in the quadratic equation $f(x) = ax^2 + bx + c$. The y-intercept is the point $(0, c)$. ➗ X-Intercepts: Also known as roots or zeros. To find them, set $f(x) = 0$ and solve for $x$. This can be done by:
• Factoring
• Using the Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Completing the Square
📊 The Discriminant: The discriminant ($b^2 - 4ac$) tells us about the nature of the roots:
• If $b^2 - 4ac > 0$: Two distinct real roots (two x-intercepts).
• If $b^2 - 4ac = 0$: One real root (one x-intercept).
• If $b^2 - 4ac < 0$: No real roots (no x-intercepts).

Practice Quiz

1. Question 1: What is the y-intercept of the quadratic function $f(x) = 2x^2 - 5x + 3$?
1. (0, -3)
2. (0, 2)
3. (0, 3)
4. (3, 0)
2. Question 2: How many x-intercepts does the quadratic function $f(x) = x^2 + 4x + 4$ have?
1. 0
2. 1
3. 2
4. 3
3. Question 3: What are the x-intercepts of the quadratic function $f(x) = x^2 - 9$?
1. (-3, 0) and (3, 0)
2. (0, -3) and (0, 3)
3. (9, 0)
4. No x-intercepts
4. Question 4: Find the x-intercepts of the function $f(x) = x^2 - 5x + 6$.
1. (1, 0) and (6, 0)
2. (-2, 0) and (-3, 0)
3. (2, 0) and (3, 0)
4. No x-intercepts
5. Question 5: What is the y-intercept of $f(x) = -3x^2 + 7x - 2$?
1. (0, 2)
2. (0, -2)
3. (2, 0)
4. (-2, 0)
6. Question 6: Which of the following quadratic functions has no real x-intercepts?
1. $f(x) = x^2 - 4x + 3$
2. $f(x) = x^2 - 6x + 9$
3. $f(x) = x^2 + 2x + 5$
4. $f(x) = x^2 - 1$
7. Question 7: What are the x-intercepts of $f(x) = 2x^2 + 5x - 3$?
1. (-3, 0) and (1/2, 0)
2. (3, 0) and (-1/2, 0)
3. (-3, 0) and (-1/2, 0)
4. (3, 0) and (1/2, 0)