Grade 9 · Mathematics
Quarter 3 · Week 1 · TEKS
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This lesson guides Grade 9 students through writing quadratic functions when given real solutions and graphs of their related equations. Students will learn to use the properties of quadratic functions to write equations in multiple forms, connecting algebraic expressions with their graphical representations. The lesson aligns with TEKS.111.39.06C and emphasizes understanding the relationship between zeros, intercepts, and the quadratic function's equation.
Students should understand the standard form of quadratic functions, the concept of zeros/roots, and basic graphing of quadratic functions, including identifying x- and y-intercepts.
Real solutions (also called zeros or roots) of a quadratic function are the x-values where the graph crosses the x-axis. These points satisfy the equation f(x) = 0. If a quadratic function has real solutions r1 and r2, then the function can be written in factored form as f(x) = a(x - r1)(x - r2), where 'a' is a nonzero constant that affects the graph's vertical stretch or compression and direction (up or down). Understanding this relationship is key to writing quadratic functions from given solutions or graphs.
When given two real solutions, students can write the quadratic function in factored form by substituting the solutions into the factors. For example, if the solutions are x = 3 and x = -2, the function can be written as f(x) = a(x - 3)(x + 2). The value of 'a' can be determined if a point on the graph (other than the zeros) is known, or assumed as 1 for basic forms. This method connects algebraic expressions directly to the zeros of the function.
Graphs of quadratic functions show the parabola's shape and intercepts. The x-intercepts correspond to the real solutions. By identifying these points on the graph, students can write the function in factored form. Additionally, the y-intercept can help find the value of 'a' if needed. For example, if the graph crosses the x-axis at x = 1 and x = 4, the function is f(x) = a(x - 1)(x - 4). If the y-intercept is (0, -4), substitute x=0 and f(x)=-4 to solve for 'a'.
A common misconception is confusing the signs in the factors: for a root r, the factor is (x - r), not (x + r). Another is assuming the value of 'a' is always 1 without checking if the graph is vertically stretched or reflected. Students may also confuse zeros with y-intercepts; zeros are x-values where the function equals zero, while the y-intercept is the value when x=0.
Graphing calculators or software can help students visualize quadratic functions and confirm their written equations. By plotting points and adjusting parameters, students see how changes in 'a', 'b', and 'c' affect the graph. This supports deeper understanding and connects symbolic and graphical representations.
Given the real solutions x = 2 and x = -3, write the quadratic function in factored form assuming a = 1.
The factors corresponding to the solutions are (x - 2) and (x + 3). Multiplying these gives the quadratic function f(x) = (x - 2)(x + 3). Since a = 1, the function is f(x) = (x - 2)(x + 3).
f(x) = (x - 2)(x + 3)
A quadratic function has real solutions at x = 1 and x = 4, and passes through the point (0, -4). Write the quadratic function in factored form and find the value of 'a'.
Start with f(x) = a(x - 1)(x - 4). Substitute the point (0, -4): -4 = a(0 - 1)(0 - 4) = a(-1)(-4) = 4a. Solve for a: a = -4/4 = -1. The function is f(x) = -1(x - 1)(x - 4).
f(x) = -1(x - 1)(x - 4)
Show students a graph of a quadratic function with clear x-intercepts and ask them what the points where the graph crosses the x-axis represent.
Students observe the graph and respond that these points are the zeros or real solutions of the quadratic function.
The points where the graph crosses the x-axis are the real solutions because at these points, the function's value is zero.
Provide students with graphs of quadratic functions and ask them to identify the real solutions from the graphs.
Students work individually or in pairs to find the x-intercepts on the graphs and write down the real solutions.
By locating the x-intercepts, students find the values of x that make the function zero, which are the real solutions needed to write the function in factored form.
Demonstrate how to write a quadratic function in factored form given real solutions, including determining the value of 'a' using a point on the graph.
Students follow along and practice writing quadratic functions from given solutions and points.
Using the real solutions r1 and r2, the function is f(x) = a(x - r1)(x - r2). To find 'a', substitute a known point (x,y) on the graph into the equation and solve for 'a'.
Present a real-world problem where students must write a quadratic function from given solutions or a graph and interpret the meaning of the solutions.
Students solve the problem by identifying solutions and writing the quadratic function, then explain the significance of the solutions in context.
Real-world problems often involve finding quadratic functions that model situations. The solutions represent meaningful values, such as times or distances, and writing the function helps analyze the situation.
Give students an exit ticket with a graph showing two real solutions and a point on the graph. Ask them to write the quadratic function in factored form and find 'a'.
Students complete the exit ticket independently, writing the quadratic function and explaining their steps.
This task assesses students' ability to identify real solutions from a graph, write the function in factored form, and determine the value of 'a' using a point.
Students work individually or in pairs to find the x-intercepts on the graphs and write down the real solutions.
Identify the real solutions from a graph where the parabola crosses the x-axis at x = -2 and x = 5.
The real solutions are x = -2 and x = 5.
Write the quadratic function in factored form given the real solutions x = 3 and x = -1, assuming a = 1.
f(x) = (x - 3)(x + 1)
Given a quadratic function with real solutions x = 0 and x = 4, and the point (1, -3) on the graph, write the quadratic function in factored form and find 'a'.
f(x) = x(x - 4)
A graph shows a parabola crossing the x-axis at x = 2 and x = 6, with a y-intercept at (0, -8). Write the quadratic function in factored form and find 'a'.
f(x) = (-2/3)(x - 2)(x - 6)
Given a quadratic function with real solutions x = -1 and x = 5, and the point (2, 12) on the graph, write the quadratic function in factored form and determine the value of 'a'.
f(x) = (-4/3)(x + 1)(x - 5)
Provide step-by-step templates for writing quadratic functions from solutions. Use graphing technology to help visualize zeros and intercepts. Offer guided notes highlighting the relationship between roots and factors.
Use visual aids and gestures to explain zeros and factors. Provide vocabulary cards for key terms like 'zero', 'root', 'factor'. Pair ELL students with peers for collaborative practice.
Challenge students to write quadratic functions from graphs with vertex form transformations. Explore how changing 'a' affects the graph's width and direction. Investigate quadratic functions with complex roots and discuss differences.
Ask students to find real-world examples of situations modeled by quadratic functions, such as projectile motion, and identify the zeros and vertex from graphs or data.
Students learned to write quadratic functions when given real solutions and graphs by identifying zeros from graphs and writing the function in factored form. They practiced finding the value of 'a' using points on the graph and connected the algebraic form to the graphical representation. This skill helps in understanding the relationship between equations and their graphs and prepares students for solving quadratic equations and modeling real-world problems.
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