CodeMathFusion

Graphing Quadratic Functions

1️⃣ Understanding the Graph of a Quadratic Function

A quadratic function is any function that can be written in the form:

\[ f(x)=ax^2+bx+c \]

  • ✔ The graph of a quadratic function is a parabola.
  • ✔ If \( a>0 \), the parabola opens upwards (U-shape).
  • ✔ If \( a<0 \), the parabola opens downwards (∩-shape).

Example: Graph \( f(x)=x^2-4x+3 \). Since \( a=1>0 \), the parabola opens upward.

2️⃣ Key Features of a Parabola

1️⃣ Vertex – The Turning Point

  • ✔ The vertex is the highest or lowest point.
  • ✔ It is given by the formula: \[ x=\frac{-b}{2a} \]
  • ✔ Substitute \( x \) back into the function to find \( y \).

Example: Find the vertex of \( f(x)=x^2-6x+5 \):

\( x=\frac{6}{2}=3 \)

Then, \( f(3)=9-18+5=-4 \). Vertex = \( (3,-4) \).

2️⃣ Axis of Symmetry

The vertical line through the vertex, given by the same formula \( x=\frac{-b}{2a} \), divides the parabola into mirror images.

Example: For \( f(x)=2x^2-4x+1 \), the vertex is \( (1,-1) \) so the axis of symmetry is \( x=1 \).

3️⃣ X-Intercepts (Roots)

The x-intercepts occur where \( f(x)=0 \). Solve the equation \( ax^2+bx+c=0 \) to find them.

Example: Find the x-intercepts of \( f(x)=x^2-5x+6 \):

\( (x-2)(x-3)=0 \)

Thus, \( x=2 \) and \( x=3 \). X-intercepts: \( (2,0) \) and \( (3,0) \).

4️⃣ Y-Intercept

The y-intercept is found by evaluating \( f(0) \).

Example: For \( f(x)=3x^2-5x+2 \), \( f(0)=2 \). Y-intercept: \( (0,2) \).

3️⃣ How to Graph a Quadratic Function

Example: Graph \( f(x)=x^2-4x+3 \):

  1. Find the vertex: \( x=\frac{-(-4)}{2(1)}=\frac{4}{2}=2 \) and \( f(2)=4-8+3=-1 \). Vertex = \( (2,-1) \).
  2. Find the x-intercepts: Factor to get \( (x-1)(x-3)=0 \); thus, \( x=1 \) and \( x=3 \). X-intercepts: \( (1,0) \) and \( (3,0) \).
  3. Find the y-intercept: \( f(0)=3 \). Y-intercept: \( (0,3) \).
  4. Plot the points and draw a smooth parabola.

Practice Questions – Graph and Analyze

  1. Find the vertex, axis of symmetry, x-intercepts, and y-intercept for: \[ f(x)=x^2-6x+8 \]
  2. Find the axis of symmetry and graph: \[ f(x)=2x^2-4x+1 \]
  3. Find the x-intercepts of: \[ f(x)=-x^2+7x-12 \]
  4. Determine the vertex and graph: \[ f(x)=x^2+4x+5 \]
  5. Find the y-intercept of: \[ f(x)=3x^2-5x+2 \]
  6. Identify if the function opens upward or downward and sketch: \[ f(x)=-2x^2+5x-3 \]
  7. Find the range of: \[ f(x)=-x^2+6x-9 \]
  8. Solve for \( x \) where \( f(x)=0 \) in: \[ f(x)=4x^2-16x+15 \]
  9. Determine the maximum or minimum value for: \[ f(x)=-3x^2+12x-4 \]
  10. Write an equation for a parabola with vertex at \( (3,2) \) and passing through \( (0,5) \).

Challenging Problems – Push Your Limits!

  1. The function \( h(t)=-5t^2+20t+30 \) models the height of a projectile. Find the maximum height.
  2. Find the equation of a quadratic function with x-intercepts \( x=2 \) and \( x=5 \) and a y-intercept of 10.
  3. A farmer is fencing a rectangular garden next to a river using 60 m of fencing (no fence is needed along the river). Find the maximum area he can enclose.
  4. If a ball is thrown upward with a velocity of 30 m/s, its height is \( h(t)=-4.9t^2+30t \). Find when the ball hits the ground.
  5. The profit function is \( P(x)=-2x^2+8x+15 \). Find the maximum profit and the number of items sold to achieve it.

7️⃣ Summary

  • ✅ A quadratic function graphs as a parabola.
  • ✅ Key features include the vertex, axis of symmetry, x-intercepts, and y-intercept.
  • ✅ Graphing quadratics helps visualize maximum and minimum values.

8️⃣ Fantastic Job!

Great work! Ready to explore the next advanced topic? Let's move on to more applications of quadratic functions!

Back to Algebra Syllabus