10.1: A Linear Equation and Its Graph (5 minutes)
In this warm-up, students revisit the graph of a linear equation and recall that an equation can tell us something about the graph that represents it and vice versa.
Here is a graph of the equation \(y = 8-2x\).
- Where do you see the 8 from the equation in the graph?
- Where do you see the -2 from the equation in the graph?
- What is the \(x\)-intercept of the graph? How does this relate to the equation?
Make sure students recall that:
- The \(y\)-intercept tells us where a graph intersects the \(y\)-axis, at which point the \(x\) value is 0. When a linear equation is written in the form \(y=mx+b\), the \(b\) tells us the \(y\)-intercept because when \(x\) is 0, \(mx\) is also 0, so \(y=b\) and \((0,b)\) is the \(y\)-intercept.
- The \(x\)-intercept tells us where a graph intersects the \(x\)-axis, at which point the \(y\) value is 0. To find where the graph intersects the \(x\)-axis, find the solution to \(0 = mx + b\).
Tell students that, when written in certain forms, quadratic expressions also tell us information about their graphs, and vice versa. We’ll explore these connections in this lesson and upcoming ones.
10.2: Revisiting Projectile Motion (15 minutes)
In the past few lessons, students have reasoned symbolically and formally about quadratic expressions. This activity ties that work back to the quadratic functions that represent situations from earlier in the unit.
Students are given two expressions that represent a familiar quadratic function. They identify these expressions by their form and justify why the two are equivalent, using what they learned about expanding factored expressions. Students also study the graph that represents the function and interpret the features of the graph in context. Students are asked to interpret the meaning of the horizontal and vertical intercepts rather than the \(x\)- and \(y\)-intercepts since these variables are not used to define the function.
Supports accessibility for: Language; Social-emotional skills
In an earlier lesson, we saw that an equation such as \(h(t) = 10 + 78t - 16t^2\) can model the height of an object thrown upward from a height of 10 feet with a vertical velocity of 78 feet per second.
- Is the expression \(10 + 78t - 16t^2\) written in standard form? Explain how you know.
- Jada said that the equation \(g(t) = (\text-16t-2)(t-5)\) also defines the same function, written in factored form. Show that Jada is correct.
- Here is a graph representing both \(g(t) = (\text-16t-2)(t-5)\) and \(h(t) = 10 + 78t - 16t^2\).
- Identify or approximate the vertical and horizontal intercepts.
- What do each of these points mean in this situation?
Some students may not recognize that the quadratic terms are the same if they identify the squared term as \(16t^2\) rather than \(\text-16t^2\). Some may ignore the subtraction sign in \(10+78t-16t^2\), or associate it with the \(78t\) rather than the \(16t^2\). Show them a simpler problem expression such as \(5-10\) and rewrite it as \(5+\text-10\). Then, ask them to rewrite the expression defining \(h\) in a similar way.
Make sure that:
- Students see that the expression \(10 + 78t - 16t^2\) is in standard form, even though it is written as \(c + bx + ax^2\). Clarify that -16 is \(a\) (the coefficient of the squared term), 78 is \(b\) (the coefficient of the linear term), and 10 is \(c\) (the constant term).
- Students can explain the equivalence of \((\text-16t-2)(t-5)\) and \(10 + 78t - 16t^2\) using a strategy from earlier lessons (for example, drawing a diagram or applying the distributive property).
Ask students if they notice any connections between the two equations and the features of the graph. Some may notice the 10 in the standard form of the equation tells us the vertical intercept is \((0,10)\). Others may predict that the subtraction of 5 has something to do with the horizontal intercepts or may not see any connections. Any observation is fine at this point, as students will look closely at equations and graphs starting in the next activity.
Design Principle(s): Maximize meta-awareness; Support sense-making
10.3: Relating Expressions and Their Graphs (15 minutes)
The goal of this activity is to uncover the connections between the \(x\)- and \(y\)-intercepts on the graph and the parameters of quadratic expressions in standard and factored form. Earlier in the unit, students learned that the zeros of a quadratic function are \(x\)-values that produce \(y\)-value of 0, and that the zeros of a function are the \(x\)-coordinates of the \(x\)-intercepts of the graph. This idea comes into focus in this activity.
Note that in the examples given here, the \(y\)-coordinate of the \(y\)-intercept is always equal to the product of the zeros, but this is not the case with all graphs representing quadratic functions. (For example, graph representing \(y=2x^2-8\) intercepts the \(x\)-axis at 2 and -2, but the \(y\)-intercept is \((0,\text-8)\).) If students make this observation, consider acknowledging that this seems to be true for these graphs and prompting students to revisit this observation in upcoming lessons (in which students will also study graphs) to see if it is always true.
Arrange students in groups of 2. Give students a few minutes of quiet time to complete the first two questions. Then, ask them to discuss their observations with a partner before completing the last question.
Here are pairs of expressions in standard and factored forms. Each pair of expressions define the same quadratic function, which can be represented with the given graph.
Identify the \(x\)-intercepts and the \(y\)-intercept of each graph.
\(x^2 + 4x + 3\)
\((x + 3)(x+1)\)
\(x^2 - 5x + 4\)
\((x - 4)(x-1)\)
\(x^2 - 9\)
\((x - 3)(x+3)\)
\(x^2 - 5x\)
\(5x - x^2\)
What do you notice about the \(x\)-intercepts, the \(y\)-intercept, and the numbers in the expressions defining each function? Make a couple of observations.
Here is an expression that models function \(p\), another quadratic function: \((x-9)(x-1)\). Predict the \(x\)-intercepts and the \(y\)-intercept of the graph that represent this function.
Are you ready for more?
Find the values of \(a\), \(p\), and \(q\) that will make \(y=a(x-p)(x-q)\) be the equation represented by the graph.
Invite students to share their observations on how the numbers in the quadratic expressions relate to the intercepts of the graphs. Then, ask them to share their predictions for the \(x\)- and \(y\)-intercepts of the graph of function \(p\), defined by \((x-9)(x-1)\). Discuss with students:
- “How did you find the \(x\)-intercept of the graph of function \(p\) without graphing? (By looking at the intercepts in other graphs with similar factors. In the examples, expressions of the form \((x-a)(x-b)\) have \((a,0)\) and \((b,0)\) for the \(x\)-intercepts.)
- “How did you find the \(y\)-intercept?” (By writing the expression in the standard form and seeing what the constant term is. By evaluating the expression at \(x=0\).)
Demonstrate graphing \(p(x)=(x-9)(x-1)\) using the technology available in your classroom. Point to the intercepts. If using Desmos, show students that you can click on the intercepts to reveal the coordinates of the points.
Remind students that earlier in the unit we learned that the \(x\)-intercepts of a graph tell us the zeros of the function, or the input values that produce an output of 0. Highlight that, because an expression in factored form can tell us about the \(x\)-intercept of the graph, this form is also handy for telling us about the zeros of the function that the expression represents.
At this point, students are just noticing that numbers in the expression have something to do with intercepts on the graph. Tell students that, in the next lesson, we will explore why they are related.
Supports accessibility for: Visual-spatial processing
In this lesson, we saw that quadratic expressions can give us clues about the intercepts of their graphs, and graphs can give us insights about the expressions they represent. Ask students:
- “If we graph \(y=(x-7)(x-3)\) and \(y=x^2-10x+21\), will we end up with the same graph? How do you know?” (Yes. The two expressions that define \(y\) are equivalent. Expanding \((x-7)(x-3)\) gives \(x^2-7x-3x+21\) or \(x^2-10x+21\).)
- “Where do you predict the intercepts of the graph will be? How do you make your predictions?” (The \(y\)-intercept will be \((0,21)\) and the \(x\)-intercepts will be \((7,0)\) and \((3,0)\). In the examples in the lesson, we saw that the numbers in the factored form give a clue about the \(x\)-intercepts, and the constant term in the standard form gives a clue about the \(y\)-intercept. Also, evaluating \(y\) when \(x=0\) gives us the \(y\)-intercept.)
- “What do the \(x\)-intercepts of a graph tell us about the quadratic function it represents?” (They tell us the zeros of the function.)
10.4: Cool-down - Making Connections (5 minutes)
Student Lesson Summary
Different forms of quadratic functions can tell us interesting information about the function’s graph. When a quadratic function is expressed in standard form, it can tell us the \(y\)-intercept of the graph representing the function. For example, the graph representing \(y=x^2 -5x + 7\) has its \(y\)-intercept \((0,7)\). This makes sense because the \(y\)-coordinate is the \(y\)-value when \(x\) is 0. Evaluating the expression at \(x=0\) gives \(y=0^2-5(0)+7\), which equals 7.
When a function is expressed in factored form, it can help us see the \(x\)-intercepts of its graph. Let’s look at the functions \(f\) given by \(f(x) = (x-4)(x-1)\) and \(g\) given by \(g(x)=(x+2)(x+6)\).
If we graph \(y = f(x)\), we see that the \(x\)-intercepts of the graph are \((1,0)\) and \((4,0)\). Notice that “1” and “4” also appear in \(f(x) = (x-4)(x-1)\), and they are subtracted from \(x\).
If we graph \(y=g(x)\), we see that the \(x\)-intercepts are at \((\text-2,0)\) and \((\text-6,0)\). Notice that “2” and “6” are also in the equation \(g(x)=(x+2)(x+6)\), but they are added to \(x\).
The connection between the factored form and the \(x\)-intercepts of the graph tells us about the zeros of the function (the input values that produce an output of 0). In the next lesson, we will further explore these connections between different forms of quadratic expressions and the graphs representing them.